Receiver, transmitter, system and method employing space-delay precoding

ABSTRACT

A receiver receives and processes a radio signal received via a frequency selective radio channel from a transmitter employing a plurality of transmit antennas. The receiver determines, based on the received signal, complex precoder coefficients and delays of respective space-delay precoders for each layer and transmit antenna at the transmitter so as to achieve a predefined property for a communication over the radio channel, each space-delay precoder modeling or defining for the associated transmit antenna a plurality of cyclic filters delaying and weighting a signal to be transmitted with the corresponding precoder delays and complex precoder coefficients, respectively, and feeds back to the transmitter the determined delays explicitly or implicitly and the determined complex precoder coefficients explicitly or implicitly, the transmitter precoding the signals to be transmitted to the receiver using the fed back delays and complex precoder coefficients.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a continuation of copending InternationalApplication No. PCT/EP2018/055678, filed Mar. 7, 2018, which isincorporated herein by reference in its entirety, and additionallyclaims priority from European Application No. EP 17197119.5, filed Oct.18, 2017, which is incorporated herein by reference in its entirety.

The present invention concerns the field of wireless communicationsystems, such as a mobile communication network. Embodiments of thepresent invention relate to wireless communication systems employingprecoding with reduced feedback, e.g., space-delay wideband MIMO(Multiple Input Multiple Output) precoding for mmWave systems

BACKGROUND OF THE INVENTION

FIG. 1 is a schematic representation of an example of a wireless network100 including a core network 102 and a radio access network 104. Theradio access network 104 may include a plurality of base stations eNB₁to eNB₅, each serving a specific area surrounding the base stationschematically represented by respective cells 106 ₁ to 106 ₅. The basestations are provided to serve users within a cell. A user may be astationary device or a mobile device. Further, the wirelesscommunication system may be accessed by mobile or stationary IoT deviceswhich connect to a base station or to a user. The mobile devices or theIoT devices may include physical devices, ground based vehicles, such asrobots or cars, aerial vehicles, such as manned or unmanned aerialvehicles (UAVs), the latter also referred to as drones, buildings andother items having embedded therein electronics, software, sensors,actuators, or the like as well as network connectivity that enable thesedevices to collect and exchange data across an existing networkinfrastructure. FIG. 1 shows an exemplary view of only five cells,however, the wireless communication system may include more such cells.FIG. 1 shows two users UE1 and UE2, also referred to as user equipment(UE), that are in cell 106 ₂ and that are served by base station eNB₂.Another user UE₃ is shown in cell 106 ₄ which is served by base stationeNB₄. The arrows 108 ₁, 108 ₂ and 108 ₃ schematically representuplink/downlink connections for transmitting data from a user UE₁, UE₂and UE₃ to the base stations eNB₂, eNB₄ or for transmitting data fromthe base stations eNB₂, eNB₄ to the users UE₁, UE₂, UE₃. Further, FIG. 1shows two IoT devices 110 ₁ and 110 ₂ in cell 106 ₄, which may bestationary or mobile devices. The IoT device 110 ₁ accesses the wirelesscommunication system via the base station eNB₄ to receive and transmitdata as schematically represented by arrow 112 ₁. The IoT device 110 ₂accesses the wireless communication system via the user UE₃ as isschematically represented by arrow 112 ₂. The respective base stationeNB₁ to eNB₅ are connected to the core network 102 and/or with eachother via respective backhaul links 114 ₁ to 114 ₅, which areschematically represented in FIG. 1 by the arrows pointing to the“core”. The core network 102 may be connected to one or more externalnetworks.

For data transmission a physical resource grid may be used. The physicalresource grid may comprise a set of resource elements to which variousphysical channels and physical signals are mapped. For example, thephysical channels may include the physical downlink and uplink sharedchannels (PDSCH, PUSCH) carrying user specific data, also referred to asdownlink and uplink payload data, the physical broadcast channel (PBCH)carrying for example a master information block (MIB) and a systeminformation block (SIB), the physical downlink and uplink controlchannels (PDCCH, PUCCH) carrying for example the downlink controlinformation (DCI), etc. For the uplink, the physical channels mayfurther include the physical random access channel (PRACH or RACH) usedby UEs for accessing the network once a UE synchronized and obtained theMIB and SIB. The physical signals may comprise reference signals (RS),synchronization signals and the like. The resource grid may comprise aframe having a certain duration in the time domain and a given bandwidthin the frequency domain. The frame may have a certain number ofsubframes of a predefined length, and each subframe may include symbols,like OFDM symbols.

The wireless communication system may operate, e.g., in accordance withthe LTE-Advanced pro standard or the 5G or NR (New Radio) standard.

The wireless communication system may be any single-tone or multicarriersystem based on frequency-division multiplexing, like the orthogonalfrequency-division multiplexing (OFDM) system, the orthogonalfrequency-division multiple access (OFDMA) system, or any otherIFFT-based signal with or without CP, e.g. DFT-s-OFDM. Other waveforms,like non-orthogonal waveforms for multiple access, e.g. filter-bankmulticarrier (FBMC), generalized frequency division multiplexing (GFDM)or universal filtered multi carrier (UFMC), may be used.

In a wireless communication system like to one depicted schematically inFIG. 1, multi-antenna techniques may be used, e.g., in accordance withLTE or NR, to improve user data rates, link reliability, cell coverageand network capacity. To support multi-stream or multi-layertransmissions, linear precoding is used in the physical layer of thecommunication system. Linear precoding is performed by a precoder matrixwhich maps layers of data to antenna ports. The precoding may be seen asa generalization of beamforming, which is a technique to spatiallydirect/focus data transmission towards an intended receiver.

In the following the downlink (DL) transmission in a mobile multipleinput multiple output communication system is considered, i.e., thecommunication link carrying data traffic from a base station (eNodeB) toa mobile user equipment (UE). Considering a base station (eNodeB) withN_(Tx) antennas and a mobile user equipment (UE), with N_(Rx) antennas,the symbols received at a particular instant of time in a DLtransmission at the UE, y∈

^(N) ^(Rx) ^(×1), may be written asy=HFs+nwhere H∈

^(N) ^(Rx) ^(×N) ^(Tx) denotes the channel matrix, F∈

^(N) ^(Tx) ^(×N) ^(s) represents the precoder matrix at the eNodeB, n∈

^(N) ^(Rx) ^(×1) is the additive noise at the receiver, s∈

^(N) ^(s) ^(×1) is the data vector transmitted by the eNodeB which hasto be decoded by the UE, and N_(s) denotes the number of data streamstransmitted. The precoder matrix to be used at the eNodeB to map thedata s∈

^(N) ^(s) ^(×1) to the N_(Tx) antenna ports is decided by solving anoptimization problem that is based on the instantaneous channelinformation H∈

^(N) ^(Rx) ^(×N) ^(Tx) . In a closed-loop mode of communication, the UEestimates the state of the channel and transmits a report, like channelstate information (CSI), to the eNodeB via a feedback channel in theuplink (the communication link carrying traffic from the UE to theeNodeB) so that the eNodeB may determine the precoding matrix (seereference [1]). There are also occasions when multiple-layertransmissions are performed without feedback from the UE to determinethe precoding matrices. Such a mode of communication is called‘open-loop’ and the eNodeB makes use of signal diversity and spatialmultiplexing to transmit information (see reference [1]).

FIG. 2 shows a block-based model of a MIMO DL transmission usingcodebook-based-precoding in accordance with LTE release 8. FIG. 2 showsschematically the base station 200, the user equipment 300 and thechannel 400, like a radio channel for a wireless data communicationbetween the base station 200 and the user equipment 300. The basestation includes an antenna array 202 having a plurality of antennas orantenna elements, and a precoder 204 receiving a data vector 206 and aprecoder matrix F from a codebook 208. The channel 400 may be describedby the channel matrix 402. The user equipment 300 receives the datavector 302 via an antenna or an antenna array 304 having a plurality ofantennas or antenna elements. A feedback channel 500 between the userequipment 300 and the base station 200 is provided for transmittingfeedback information.

In the case of an implicit feedback, the CSI transmitted by the UE 300over the feedback channel 500 includes the rank index (RI), theprecoding matrix index (PMI) and the channel quality index (CQI)allowing, at the eNodeB 200, deciding the precoding matrix, and themodulation order and coding scheme (MCS) of the symbols to betransmitted. The PMI and the RI are used to determine the precodingmatrix from a predefined set of matrices Ω called ‘codebook’ 208. Thecodebook 208, e.g., in accordance with LTE, may be a look-up table withmatrices in each entry of the table, and the PMI and RI from the UEdecide from which row and column of the table the precoder matrix to beused is obtained.

With explicit CSI feedback, no codebook is used to determine theprecoder. The coefficients of the precoder matrix are transmittedexplicitly by the UE. Alternatively, the coefficients of theinstantaneous channel matrix may be transmitted, from which the precoderis determined by the eNodeB.

The design and optimization of the precoder 204 and the codebook 208 maybe performed for eNodeBs equipped with 1-dimensional Uniform LinearArrays (ULAs) or 2-dimensional Uniform Planar Arrays (UPAs) having afixed down-tilt. These antenna arrays 202 allow controlling the radiowave in the horizontal (azimuth) direction so that azimuth-onlybeamforming at the eNodeB 200 is possible. In accordance with otherexamples, the design of the codebook 208 is extended to support UPAs fortransmit beamforming on both vertical (elevation) and horizontal(azimuth) directions, which is also referred to as full-dimension (FD)MIMO (see reference [2]). The codebook 208, e.g., in the case of massiveantenna arrays such as FD-MIMO, may be a set of beamforming weights thatforms spatially separated electromagnetic transmit/receive beams usingthe array response vectors of the array. The beamforming weights (or the‘array steering vectors’) of the array are amplitude gains and phaseadjustments that are applied to the signal fed to the antennas (or thesignal received from the antennas) to transmit (or obtain) a radiationtowards (or from) a particular direction. The components of the precodermatrix are obtained from the codebook of the array, and the PMI and theRI are used to ‘read’ the codebook and obtain the precoder. The arraysteering vectors may be described by the columns of a 2-D DiscreteFourier Transform (DFT) matrix (see reference [3]).

The frequency-domain precoder matrices used in the Type-I and Type-IICSI reporting schemes in 3GPP New Radio Release 15 have a dual-stagestructure: F(s)=F₁F₂(s), s=0 . . . , S−1 (see reference [7]), where Sdenotes the number of subbands/subcarriers or physical resource blocks(PRB). The matrix F₁ is a wide-band matrix, independent on index s, andcontains PU beamforming vectors s_(u) ^(p)∈

^(C×1), p=1, . . . , P selected out of a DFT codebook matrix,

${{F_{1}(s)} = {\begin{bmatrix}s_{1}^{1} & \ldots & s_{u}^{1} & \ldots & s_{U}^{1} & \ldots & \; & \; & 0 & \; & \; \\\; & \; & \vdots & \; & \; & \ddots & \; & \; & \vdots & \; & \mspace{11mu} \\\; & \; & 0 & \; & \; & \ldots & s_{1}^{P} & \ldots & s_{u}^{P} & \ldots & s_{U}^{P}\end{bmatrix} \in {\mathbb{C}}^{{AP} \times {UP}}}},$where A denotes the number of transmit antennas per polarization, and Pdenotes the number of antenna polarizations, and U is the number ofbeamforming vectors per polarization. For co-polarized antenna arrays,P=1, whereas for dual-polarized antenna arrays, P=2. Moreover, fordual-polarized antenna arrays, the u-th beam vectors s_(u) ¹=s_(u) ², ∀uare identical for both polarizations. The matrix F₂ (s) is aselection/combining/co-phasing matrix that selects/combines/co-phase thebeams defined in F₁ for each subband/subcarrier or physical resourceblock (PRB) s. It is noted that multiple antenna elements oriented indifferent directions may be placed at each position in an array antennato make use of the polarization diversity while transmitting/receiving asignal. The orientation of the antenna element in many cases is the sameas the polarization angle the antenna responds to and, hence, the term‘antenna polarization’ and ‘antenna orientation’ are usedinterchangeably across literature. In this specification, the term‘orientation’ is used when referring to antennas to avoid confusing withthe polarization of a transmitted or a received wavefront.

For a rank-1 transmission and Type-I reporting, F₂ (s) is given fordual-polarized antenna arrays (P=2) by [7]

${{F_{2}(s)} = {\begin{bmatrix}e_{u} \\{e^{j\;\delta_{1}}e_{u}}\end{bmatrix} \in {\mathbb{C}}^{{U \cdot 2} \times 1}}},$where e_(u)∈

^(U×1), u=1, 2, . . . , U contains zeros at all positions except theu_(th) position. Such a definition of e_(u) selects the u_(th) vectorfor each polarization and combines them across different polarizations.Furthermore, δ₁ is a quantized phase adjustment for the secondpolarization.

For a rank-1 transmission and Type-II reporting, F₂ (s) is given fordual-polarized antenna arrays (P=2) by [7]

${F_{2}(s)} = {\begin{bmatrix}{e^{j\;\delta_{1}}p_{1}} \\{e^{j\;\delta_{2U}}p_{2U}}\end{bmatrix} \in {\mathbb{C}}^{{U \cdot 2} \times 1}}$where the quantized values p_(u) and δ_(u), u=1, 2, . . . , 2U are theamplitude and phase combing coefficients, respectively.

For rank-R transmission, F₂ (s) contains R vectors, where the entries ofeach vector are chosen to combine single or multiple beams within eachpolarization and/or combining them across different polarizations.

SUMMARY

According to an embodiment, a receiver may be configured to

receive and process a radio signal received via a frequency selectiveradio channel from a transmitter employing a plurality of transmitantennas,

determine, based on the received signal, complex precoder coefficientsand delays of respective space-delay precoders for each layer andtransmit beam at the transmitter so as to achieve a predefined propertyfor a communication over the radio channel, wherein each space-delayprecoder may have: a double-stage precoding structure, the double-stageprecoding structure including a beamforming matrix that contains PUspatial beams, U=total number of beams, and P=number of polarizations,where P=1 for co-polarized antenna arrays at the transmitter and P=2dual-polarized antenna arrays at the transmitter, wherein thedouble-stage precoding structure further includes a space-delay-domaincombining coefficient vector or matrix including complex delay-domaincombining-coefficients associated with the beams and the delays,feed back to the transmitter the determined delays and the determinedcomplex precoder coefficients, the complex precoder coefficientsincluding the complex delay-domain combining-coefficients,wherein the feedback includes a precoding matrix identifier, PMI, thePMI indicating a number of indices of the respective spatial beams ofthe radio signal, the respective complex delay-domaincombining-coefficients, and a number of indices of the delays associatedwith respective column vectors of a codebook matrix.

According to another embodiment, a transmitter may have: an antennaarray having a plurality of antennas for a wireless communication withone or more receivers; and a precoder connected to the antenna array,the precoder to apply a set of beamforming weights to one or moreantennas of the antenna array to form, by the antenna array, one or moretransmit beams, wherein the transmitter is configured to determine thebeamforming weights responsive to a feedback received from a receiver,the feedback indicating delays and complex precoder coefficients, theindicated delays and complex precoder coefficients obtained based onrespective space-delay precoders for each layer and transmit beam at thetransmitter so as to achieve a predefined property for a communicationover a radio channel to the receiver, the complex precoder coefficientsincluding complex delay-domain combining-coefficients, and wherein eachspace-delay precoder may have: a double-stage precoding structure, thedouble-stage precoding structure including a beamforming matrix thatcontains PU spatial beams, U=total number of beams, and P=number ofpolarizations, where P=1 for co-polarized antenna arrays at thetransmitter and P=2 dual-polarized antenna arrays at the transmitter,wherein the double-stage precoding structure further includes aspace-delay-domain combining coefficient vector or matrix including thecomplex delay-domain combining-coefficients associated with the beamsand the delays, and wherein the feedback includes a precoding matrixidentifier, PMI, the PMI indicating a number of indices of therespective spatial beams of the radio signal, the respective complexdelay-domain combining-coefficients, and a number of indices of thedelays associated with respective column vectors of a codebook matrix.

According to another embodiment, a wireless communication network mayhave: at least one inventive receiver, and at least one inventivereceiver.

According to another embodiment, a method may have the steps of:receiving and processing a radio signal received via a frequencyselective radio channel from a transmitter employing a plurality oftransmit antennas, determining, based on the received signal, complexprecoder coefficients and delays of respective space-delay precoders foreach layer and transmit beam at the transmitter so as to achieve apredefined property for a communication over the radio channel, whereineach space-delay precoder may have a double-stage precoding structure,the double-stage precoding structure including a beamforming matrix thatcontains PU spatial beams, U=total number of beams, and P=number ofpolarizations, where P=1 for co-polarized antenna arrays at thetransmitter and P=2 dual-polarized antenna arrays at the transmitter,wherein the double-stage precoding structure further includes aspace-delay-domain combining coefficient vector or matrix includingcomplex delay-domain combining-coefficients associated with the beamsand the delays, and feeding back to the transmitter the determineddelays and the determined complex precoder coefficients, the complexprecoder coefficients including the complex delay-domaincombining-coefficients, wherein the feedback includes a precoding matrixidentifier, PMI, the PMI indicating a number of indices of therespective spatial beams of the radio signal, the respective complexdelay-domain combining-coefficients, and a number of indices of the ofthe delays associated with respective column vectors of a codebookmatrix.

According to another embodiment, a method for forming one or more beamsfor a wireless communication among a transmitter and one or morereceivers may have the step of: applying a set of beamforming weights toone or more antennas of an antenna array to form the beam, the beamcomprising a transmit beam, wherein the beamforming weights aredetermined responsive to a feedback received from a receiver, thefeedback indicating delays and complex precoder coefficients, theindicated delays and complex precoder coefficients obtained based onrespective space-delay precoders for each layer and transmit beam at thetransmitter so as to achieve a predefined property for a communicationover a radio channel to the receiver, the complex precoder coefficientsincluding complex delay-domain combining-coefficients, and wherein eachspace-delay precoder may have a double-stage precoding structure, thedouble-stage precoding structure including a beamforming matrix thatcontains PU spatial beams, U=total number of beams, and P=number ofpolarizations, where P=1 for co-polarized antenna arrays at thetransmitter and P=2 dual-polarized antenna arrays at the transmitter,wherein the double-stage precoding structure further includes aspace-delay-domain combining coefficient vector or matrix including thecomplex delay-domain combining-coefficients associated with the beamsand the delays, and wherein the feedback includes wherein the feedbackincludes a precoding matrix identifier, PMI, the PMI indicating a numberof indices of the respective spatial beams of the radio signal, therespective complex delay-domain combining-coefficients, and a number ofindices of the of the delays associated with respective column vectorsof a codebook matrix.

According to another embodiment, a non-transitory computer programproduct may have a computer readable medium storing instructions which,when executed on a computer, perform the inventive methods.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention will be detailed subsequentlyreferring to the appended drawings, in which:

FIG. 1 shows a schematic representation of an example of a wirelesscommunication system;

FIG. 2 shows a block-based model of a MIMO communication system usingimplicit CSI feedback;

FIG. 3 shows a block diagram of a MIMO system in accordance withembodiments of the inventive approach;

FIG. 4 shows a block diagram of a MIMO system in accordance with furtherembodiments of the inventive approach;

FIG. 5 illustrates the L delay indices for the u-th beam centered aroundthe mean delay index b_(u,1);

FIG. 6 illustrates illustrate possible locations (see FIG. 6(a) and FIG.6(b)) for the mean delay of FIG. 5 lying at the beginning and/or at theend of the sampling grid;

FIG. 7 illustrates the C delay indices centered around two mean delayindices b_(u,1) and b_(u,2) for the u-th beam;

FIG. 8 illustrates the calculating of the complex coefficients of the(2U−1) beams with respect to a reference beam for the mean delayb_(u,{circumflex over (l)}); and

FIG. 9 illustrates an example of a computer system on which units ormodules as well as the steps of the methods described in accordance withthe inventive approach may execute.

DETAILED DESCRIPTION OF THE INVENTION

In the following, embodiments of the present invention will be describedin further detail with reference to the enclosed drawings in whichelements having the same or similar function are referenced by the samereference signs.

Embodiments of the present invention provide a receiver which receivesand processes a radio signal received via a frequency selective radiochannel from a transmitter employing a plurality of transmit antennas.The receiver determines, based on the received signal, complex precodercoefficients and delays of respective space-delay precoders for eachlayer and transmit antenna at the transmitter so as to achieve apredefined property for a communication over the radio channel, eachspace-delay precoder modeling or defining for the associated transmitantenna a plurality of cyclic filters delaying and weighting a signal tobe transmitted with the corresponding precoder delays and complexprecoder coefficients, respectively, and feeds back to the transmitterthe determined delays explicitly or implicitly and the determinedcomplex precoder coefficients explicitly or implicitly, the transmitterprecoding the signals to be transmitted to the receiver using the fedback delays and complex precoder coefficients.

Further embodiments of the present invention provide a transmitterhaving an antenna array having a plurality of antennas for a wirelesscommunication with one or more receivers, and a precoder connected tothe antenna array, the precoder to apply a set of beamforming weights toone or more antennas of the antenna array to form, by the antenna array,one or more transmit beams. The transmitter determines the beamformingweights responsive to a feedback received from a receiver, the feedbackindicating delays explicitly or implicitly and complex precodercoefficients explicitly or implicitly, the indicated delays and complexprecoder coefficients obtained based on respective space-delay precodersfor each layer and transmit antenna at the transmitter so as to achievea predefined property for a communication over a radio channel to thereceiver, each space-delay precoder modeling or defining for theassociated transmit antenna a plurality of cyclic filters delaying andweighting a signal to be transmitted with the corresponding precoderdelays and complex precoder coefficients, respectively.

As has been described above, conventionally, precoding is performed persubcarrier or per subband, a subband including multiple adjacentsubcarriers, in OFDM-based systems. Due to the large number ofsubcarriers/subbands, transmitting a single PMI/RI persubcarrier/subband to the gNB leads to a prohibitively large feedbackoverhead. The problem of such a large feedback overhead is addressed inconventional OFDM systems, which precode in the frequency domain persub-carrier or per subband, as follows. As fading gains are highlycorrelated across multiple adjacent subcarriers, a single precodingmatrix may be calculated for a number of subcarriers, i.e., per subband,which may result in a reduced feedback overhead compared to the casewhen calculating a single precoding matrix per subcarrier.

However, in situations, in which the number of subcarriers/subbands ismuch larger than the number of non-zero channel impulse responsecoefficients, precoding in the time domain may be beneficial both interms of computational complexity and feedback overhead.

Therefore, instead of precoding per subcarrier/subband, per delayprecoding is performed in accordance with the inventive approach. Inaccordance with embodiments, the inventive approach employs a novelspace-delay precoder with a reduction in feedback compared to thesubcarrier/subband precoding and with higher mutual information or rateetc. In accordance with embodiments of the present invention a precodingand feedback scheme for single and/or multi-carrier MIMO communicationsystems is provided which, in addition to the feedback parametersdescribed in 3GPP Rel. 10 (see reference [4]) like PMI, RI and CQI,provides additional feedback parameters such as tap delays for thesignal precoder at the transmitter. The inventive feedback scheme allowsfor direction and delay-based beamforming/precoding with an enhancedperformance in terms of mutual information or rate etc., compared to thestate-of-the art beamforming/precoding schemes discussed until 3GPP LTERel 14 (see reference [5]).

In accordance with embodiments of the present invention the MIMOcommunication system may be operating at mmWave frequencies. At mmWavefrequencies, the communication channels are sparse and the energy of themulti-path components is concentrated in few channel clusters or channeltaps, and a number of rays are associated with each cluster. Eachchannel cluster or channel-tap may correspond to a different delay andspatial direction. Thus, the number of dominant channel clusters orchannel taps is typically much smaller than the number of subcarriers.Therefore, in systems operating at mmWave frequencies space-delayprecoding is beneficial in terms of complexity and feedback overheadcompared to conventional frequency-domain subcarrier-based orsubband-based precoding. In accordance with the inventive approach,additional tap-delay information corresponding to dominant channelcluster directions may be exploited and fed back to the gNB. Utilizingthe additional delay information of the cluster directions in designingthe precoder may lead to an enhanced system performance in terms ofmutual information or rate etc., due to the additional degrees offreedom considered.

The present invention is also applicable to a MIMO communication systemoperating at sub-6 GHz frequencies.

In accordance with embodiments the receiver is configured to feed backthe delays of the space-delay precoder implicitly using a delayidentifier including indices associated with respective column vectorsof a frequency-domain codebook matrix used at the transmitter.

In accordance with embodiments the space-delay precoder is representedin the frequency domain, and wherein the receiver is configured toexplicitly or implicitly feed back the delays of the space-delayprecoder.

In accordance with embodiments the implicit delay feedback includes oneor more delay identifiers, DI, each delay identifier including a set ofL indices which are associated with column vectors of a frequency-domaincodebook matrix D, L=total number of delays.

In accordance with embodiments the size of the codebook matrix D isflexibly designed based on the resolution of the delays that may beused.

In accordance with embodiments

-   -   the delays, τ(l)∈        , ∀l, are discretized and are given by elements of a set        =[0, . . . , SO_(f)−1], and each value in        is associated to a column vector of the frequency-domain        codebook matrix D, with l=0, 1, . . . , L, S=total number of        subcarriers, or subbands, or physical resource blocks,    -   wherein the frequency-domain codebook matrix D is an oversampled        codebook DFT-matrix D=[d₀, d₁, . . . , d_(SO) _(f) ⁻¹], where

${d_{i} = {\begin{bmatrix}1 & e^{\frac{{- j}\; 2\pi\; i}{O_{f}S}} & \ldots & e^{\frac{{- j}\; 2\pi\;{i{({S - 1})}}}{O_{f}S}}\end{bmatrix}^{T} \in {\mathbb{C}}^{S \times 1}}},$i∈

, j=√{square root over (−1)} with O_(f) being the oversampling factor ofthe frequency-domain codebook DFT-matrix.

In accordance with embodiments the receiver is configured to receivefrom the transmitter the oversampling factor O_(f).

In accordance with embodiments a DI is associated with a spatial beam,and the feedback includes PU DIs for PU spatial beams, U=total number ofbeams, P=number of polarizations, where P=1 for co-polarized antennaarrays at the transmitter and P=2 dual-polarized antenna arrays at thetransmitter.

In accordance with embodiments

-   -   the precoder comprises a double-stage precoding structure, the        double-stage precoding structure including a beamforming matrix        that contains PU spatial beams, U=total number of beams, and        P=number of polarizations, where P=1 for co-polarized antenna        arrays at the transmitter and P=2 dual-polarized antenna arrays        at the transmitter,    -   (i) in case of identical delays for all PU beams, the feedback        includes one delay identifier, 1 DI, for the PU beams, or    -   (ii) in case of polarization-dependent and beam-dependent        delays, the feedback includes PU delay identifiers, PU DIs, for        the PU beams, each DI containing indices for the delays        associated with a single spatial beam, or    -   (iii) in case of polarization-independent and beam-dependent        delays, the feedback includes U delay identifiers, U DIs, for        the PU beams, or    -   (iv) in case of polarization-dependent and beam-independent        delays, the feedback includes P delay identifiers, P DIs, for        the PU beams, or

In accordance with embodiments the number of indices in the DIs isidentical or different with respect to the spatial beams.

In accordance with embodiments, d delay indices out of L delay indicesin a delay identifier, DI, associated with a u-th spatial beam, areidentical to the delay indices of DIs associated with one or more otherspatial beams, then the DI of the u-th spatial beam contains L-d indicesinstead of L indices.

In accordance with embodiments, in addition to beam-specific DIs thatcontain indices for specific spatial beams, a DI common to X (X=1 . . .PU) spatial beams may be used to denote indices common to X spatialbeams. Such multiple common DIs may become relevant when there aremultiple sets of identical delays among DIs of different spatial beams.

In accordance with embodiments, a DI configuration may be signaled fromthe transmitter to the receiver. A DI configuration may contain, forexample, information about

-   -   total number of indices per beam-specific DI, or    -   number of common DIs, number of indices per common DI.

In accordance with embodiments, in case the delays associated with aspatial beam are within a predefined window around a single mean delay,the delay identifier for the spatial beam includes only a single indexassociated with the mean delay.

In accordance with embodiments the receiver is configured to receivefrom the transmitter the window parameter specifying the predefinedwindow-size.

In accordance with embodiments, in case of PU beams, the feedbackincludes PU DIs for the PU beams, with each DI containing only a singleindex.

In accordance with embodiments the feedback includes a single ormultiple DIs for the spatial beams, with each DI containing a single ormultiple indices, and each index is associated with a specific meandelay of the beam.

In accordance with embodiments the PU spatial beams have the same ordifferent mean delays.

In accordance with embodiments the L complex delay-domaincombining-coefficients of the u-th spatial beam associated with acertain mean delay index are used to calculate the complexcombining-coefficients of the remaining or other PU−1 beams for thecertain mean delay index.

In accordance with embodiments the complex coefficients for theremaining 2U−1 beams corresponding to the mean delay indexb_(u,{circumflex over (l)}) of the u-th beam are given by{circumflex over (K)} ₂=[e _(1,u) . . . e _(g,u) . . . e _(2U−1,u)]^(T)⊗K _(2,u)∈

^(2U×L)where e_(g,u) is the scalar complex coefficient associated with the g-thbeam (g≠u) and K_(2,u) ∈

^(1×L) contains L delay-combining coefficients associated with the u-thbeam and mean delay index b_(u,{circumflex over (l)}).

In accordance with embodiments the feedback includes a set of indices,like a precoding matrix identifier, PMI, the set of indices comprising afirst number of indices indicating respective spatial beams of the radiosignal, a second number of indices indicating the respective complexdelay-domain combining-coefficients, and a third number of indicesassociated to the delays contained in the delay identifier(s).

In accordance with embodiments

-   -   the receiver is configured to feed back the delays of the        space-delay precoder explicitly by    -   (i) setting a reference delay to all antennas or beams, the L−1        delay differences with respect to the reference delay are fed        back to the transmitter, or    -   (ii) setting a reference delay per antenna or beam, the L−1        delay differences per antenna or beam with respect to the        reference delay per antenna or beam are fed back to the        transmitter; or    -   the receiver is configured to feed back the delays of the        space-delay precoder implicitly by    -   (i) setting a reference delay to all antennas or beams, L−1        indices associated with the L−1 delay differences with respect        to the reference delay are fed back, or    -   (ii) setting a reference delay per antenna or beam, L−1 indices        per antenna or beam associated with the L−1 delay differences        per antenna or beam with respect to the reference delay per        antenna or beam are fed back to the transmitter.

In accordance with embodiments the delays τ_(n,r) (l) areantenna-specific and layer-specific or non-antenna-specific andnon-layer-specific. In case of antenna-specific and layer-specificdelays τ_(n,r) (l) the l-th delay τ_(n,r) (l) of the n-th transmitantenna, r-th layer, is different to the l-th delay τ_(k,p) (l) of thek-th transmit antenna, p-th layer, i.e., τ_(n,r) (l)≠τ_(k,r) (l), ∀n, k,l, r, n≠k and τ_(n,r) (l)≠τ_(n,p) (l), ∀n, l, r, p, r≠p. In case ofnon-antenna-specific and non-layer-specific delays τ_(n,r) (l) the l-thdelay τ_(n,r) (l) of the n-th transmit antenna, r-th layer, is identicalto the l-th delay τ_(k,p) (l) of the k-th transmit antenna, p-th layer,i.e., τ_(n,r) (l)=τ_(k,p) (l), ∀n, k, l, r, p.

In accordance with embodiments, in case of antenna-specific andlayer-specific delays and explicit feedback of the complex precodercoefficients,

-   -   in case of an explicit feedback of the delays, the feedback        includes or the total feedback amounts to N·L·R complex precoder        coefficients and N·L·R delays, and    -   in case of an implicit feedback the delays, the feedback        includes or the total feedback amounts to N·L·R complex precoder        coefficients and L·R delay identifiers,    -   where N denotes the number of transmit antennas, L denotes the        number of delays per layer and per antenna, and R denotes the        number of layers.

In accordance with embodiments, in case of non-antenna-specific andnon-layer-specific delays and explicit feedback of the complex precodercoefficients,

-   -   in case of an explicit feedback of the delays, the feedback        includes or the total feedback amounts to N·L·R complex precoder        coefficients and L delays, the L delays being identical to all N        transmit antennas and R layers, and    -   in case of an implicit feedback of the delays, the feedback        includes N·L·R complex precoder coefficients and 1 delay        identifier that specifies L delays, wherein the delays specified        in the delay identifier are the delays of the precoder taps        identical to all N transmit antennas and R layers.

In accordance with embodiments, in case of antenna-specific andlayer-specific delays and implicit feedback of the complex precodercoefficients, the complex precoder coefficients per delay and per layerare based on one or more codebooks, and the feedback specifies matrices(PMIs) of complex precoder coefficients associated with the N transmitantennas, L delays and R layers,

-   -   in case of an explicit feedback of the delays, the feedback        includes or the total feedback amounts to L·R precoding matrix        identifiers (PMIs) and N·L·R delays, and    -   in case of an implicit feedback of the delays, the feedback        includes or the total feedback amounts to L·R precoding matrix        identifiers (PMIs) and L·R delay identifiers.

In accordance with embodiments, in case of non-antenna-specific andnon-layer-specific delays and implicit feedback of the complex precodercoefficients, the complex precoder coefficients per delay and per layerare based on one or more codebooks, and the feedback specifies matrices(PMIs) of complex precoder coefficients associated with the N transmitantennas, L delays and R layers,

-   -   in case of an explicit feedback of the delays, the feedback        includes or the total feedback amounts to L·R precoding matrix        identifiers (PMIs) and L delays, and    -   in case of an implicit feedback of the delays, the feedback        includes or the total feedback amounts to L·R precoding matrix        identifiers (PMIs) and 1 delay identifier.

In accordance with embodiments, the codebook based scheme employs aprecoder matrix per layer identical for all delays.

In accordance with embodiments, the precoder comprises a multi-stagestructure, e.g., a dual-stage structure or a triple-stage structure. Themulti-stage structure may comprise a beam-set matrix and at least onecombination vector or combination matrix including complex combiningcoefficients per delay and per layer for the N transmit antennas, and avector of delays, wherein the feedback further identifies, per delay,the complex combining coefficients explicitly or implicitly using avector indicator, so that the feedback or the total feedback furtherincludes the complex combining coefficients, when explicitly signalingthe complex combining coefficients, or L·R vector indicators, whenimplicitly signaling the complex combining coefficients.

In accordance with embodiments the complex precoder coefficients perdelay and per layer are based on one or more non-polarimetric codebooksor polarimetric codebooks. In case of polarimetric codebooks the complexprecoder coefficients per delay and per layer include:

-   -   first complex precoder coefficients per delay and layer        associated with a first polarization of a transmitted/incident        wavefront, e.g., a horizontal polarization, for all antennas of        a first orientation, and    -   second complex precoder coefficients per delay and layer        associated with a second polarization of a transmitted/incident        wavefront, e.g., a vertical polarization, for all antennas of        the first orientation, and    -   third complex precoder coefficients per delay and layer        associated with the first polarization of a transmitted/incident        wavefront, e.g., the horizontal polarization, for all antennas        of a second orientation, and    -   fourth complex precoder coefficients per delay and layer        associated with the second polarization of a        transmitted/incident wavefront, e.g., the vertical polarization,        for all antennas of the second orientation.

The feedback includes respective matrix identifiers for matrices ofcomplex precoder coefficients per delay and per layer associated withthe first polarization and the first antenna orientation, and the secondpolarization and the first antenna orientation, and the secondpolarization and the first antenna orientation, and the secondpolarization and the second antenna orientation, respectively.

The present invention may be applied to single carrier or multi-carrierwireless communication systems based on frequency division multiplexingsuch as OFDM, discrete Fourier transform spread OFDM (DFT-s-OFDM), etc.The following description of embodiments is based on an OFDM systemmodel for a multi-carrier MIMO system with N transmit antennas and Mreceive antennas. The frequency-selective channel h_(m,n) between then_(th) Tx antenna and the m_(th) Rx antenna comprises Q path components,h _(m,n)=[h _(m,n)(0) . . . h _(m,n)(u) . . . h _(m,n)(Q−1)]^(T)∈

^(Q)

The transmitted data is organized in transmission blocks, where eachblock b∈

^(SR) of length SR is linearly precoded with a precoding matrix K∈

^(NS×NR) with S being the number of subcarriers. As a result, R datalayers are transmitted per block resulting in a rank-R transmission.

Assuming a cyclic-prefix (CP) transmission, the CP being at least oflength (Q−1), the received signal vector (after CP removal) at the UEmay be written asy=HKb+n∈

^(MS)where H denotes a block-circulant MIMO channel matrix

${H = {\begin{bmatrix}H_{1,1} & H_{1,2} & \ldots & H_{1,N} \\H_{2,1} & H_{2,2} & \ldots & H_{2,N} \\\vdots & \vdots & \vdots & \vdots \\H_{M,1} & H_{M,2} & \ldots & H_{M,N}\end{bmatrix} \in {\mathbb{C}}^{{MS} \times {NS}}}},$H_(m,n) is the S×S sized circulant matrix of link (m, n) with [h_(m,n)0_(S−Q) ^(T)]^(T)∈

^(S) on its first column and n is the noise.

The precoder matrix for a rank-1 transmission is given by

${K = {\begin{bmatrix}K_{1,1} \\K_{2,1} \\\vdots \\K_{{N,1}\;}\end{bmatrix} \in {\mathbb{C}}^{{NS} \times N}}},$the precoder matrix for a rank-R transmission is given by

$K = {\begin{bmatrix}K_{1,1} & K_{1,2} & \ldots & K_{1,R} \\K_{2,1} & K_{2,2} & \ldots & K_{2,R} \\\vdots & \vdots & \vdots & \vdots \\K_{N,1} & K_{N,2} & \ldots & K_{N,R}\end{bmatrix} \in {\mathbb{C}}^{{NS} \times {SR}}}$with K_(n,r) being the circulant precoder matrix of size S×S.

The frequency-domain representation of the block-circulant MIMO channelmatrix and the precoder matrix is given by H=D_(N)HD_(M) ^(H) andK=D_(N)KD_(N) ^(H), respectively, where D_(N)=I_(N)⊗D, with D being theDFT-matrix of size S.

The MIMO channel matrix in the frequency domain is given by

$\overset{\_}{H} = {\begin{bmatrix}{\overset{\_}{H}}_{1,1} & {\overset{\_}{H}}_{1,2} & \ldots & {\overset{\_}{H}}_{1,N} \\{\overset{\_}{H}}_{2,1} & {\overset{\_}{H}}_{2,2} & \ldots & {\overset{\_}{H}}_{2,N} \\\vdots & \vdots & \vdots & \vdots \\{\overset{\_}{H}}_{M,1} & {\overset{\_}{H}}_{M,2} & \ldots & {\overset{\_}{H}}_{M,N}\end{bmatrix} \in {\mathbb{C}}^{{MS} \times {NS}}}$where H _(m,n) is a diagonal matrix with channel coefficients H_(m,n)(s) of all subcarriers on the main diagonalH _(m,n)=diag{ H _(m,n)(1) . . . H _(m,n)(s) . . . H _(m,n)(S)}.

The precoder matrix in the frequency domain for the r-th layer is givenbyK _(r)=[ K _(1,r) ^(T) , . . . ,K _(n,r) ^(T) , . . . ,K _(N,r)^(T)]^(T)where K _(n,r)=diag{K _(n,r)(1), . . . , K _(n,r)(s), . . . , K_(n,r)(S)} is a diagonal matrix that consists of precoder coefficientsof all subcarriers on the main diagonal.

By rearranging, the MIMO channel matrix associated with subcarrier s, is

${\overset{\_}{H}(s)} = {\begin{bmatrix}{{\overset{\_}{H}}_{1,1}(s)} & {{\overset{\_}{H}}_{1,2}(s)} & \ldots & {{\overset{\_}{H}}_{1,N}(s)} \\{{\overset{\_}{H}}_{2,1}(s)} & {{\overset{\_}{H}}_{2,2}(s)} & \ldots & {{\overset{\_}{H}}_{2,N}(s)} \\\vdots & \vdots & \vdots & \vdots \\{{\overset{\_}{H}}_{M,1}(s)} & {{\overset{\_}{H}}_{M,2}(s)} & \ldots & {{\overset{\_}{H}}_{M,N}(s)}\end{bmatrix} \in {\mathbb{C}}^{M \times N}}$

The precoder matrices for a rank-1 transmission associated withsubcarrier s are

${{{\overset{\_}{K}}_{1}(s)} = {\begin{bmatrix}{{\overset{\_}{K}}_{1,1}(s)} \\{{\overset{\_}{K}}_{2,1}(s)} \\\vdots \\{{\overset{\_}{K}}_{N,1}(s)}\end{bmatrix} \in {\mathbb{C}}^{N}}},$the precoder matrices for a rank-R transmission associated withsubcarrier s are

${\overset{\_}{K}(s)} = {\begin{bmatrix}{{\overset{\_}{K}}_{1,1}(s)} & {{\overset{\_}{K}}_{1,2}(s)} & \ldots & {{\overset{\_}{K}}_{1,R}(s)} \\{{\overset{\_}{K}}_{2,1}(s)} & {{\overset{\_}{K}}_{2,2}(s)} & \ldots & {{\overset{\_}{K}}_{2,R}(s)} \\\vdots & \vdots & \vdots & \vdots \\{{\overset{\_}{K}}_{N,1}(s)} & {{\overset{\_}{K}}_{N,2}(s)} & \ldots & {{\overset{\_}{K}}_{N,R}(s)}\end{bmatrix} \in {\mathbb{C}}^{N \times R}}$

FIG. 3 shows a block diagram of a MIMO system in accordance withembodiments of the inventive approach. Those elements of the MIMO systemcorresponding to elements described above with reference to FIG. 2 haveassigned thereto the same reference signs. The user equipment 300receives at the antenna or the antenna array 304 the radio signal fromthe channel 400. After removing the cyclic prefix, as is indicated at306, the user equipment 300 processes the received signal to obtain thedata vector 302. In accordance with embodiments of the presentinvention, the received signal is processed to determine, as isindicated at 308, and provide, as is indicated at 310, complex precodercoefficients and delays of respective space-delay precoders for eachlayer and transmit antenna at the base station 200 so as to achieve apredefined property for a communication over the radio channel. Forexample, at 308, the complex coefficients and the delays of thespace-delay precoder (see equation (1) below) may be optimized at the UE300 to achieve a predefined property for a communication over the radiochannel, e.g., by maximizing a cost function such as mutual informationor rate based on long- and short-term channel state information, as isdescribed in more detail below. The optimized precoder taps and delaysare fed back to the gNB 200 over the feedback channel 500 via implicitor explicit feedback schemes or a combination of both. Embodiments offeedback schemes for polarimetric and non-polarimetric cases aredescribed in more detail below. In accordance with embodiments thefeedback may include further parameters, e.g., CQI and RI as also usedin conventional approaches.

FIG. 4 shows a block diagram of a MIMO system in accordance with otherembodiments of the inventive approach. Those elements of the MIMO systemcorresponding to elements described above with reference to FIG. 2 orFIG. 3 have assigned thereto the same reference signs. At the basestation 200 also the waveform modulator 212 prior to adding the cyclicprefix 210 is indicated. The user equipment 300 receives at the antennaor the antenna array 304 the radio signal from the channel 400. Afterremoving the cyclic prefix, as is indicated at 306 and waveformdemodulation 312, the user equipment 300 processes the received signalto obtain the data vector 302. In accordance with embodiments of thepresent invention, the received signal is processed to determine, as isindicated at 308, and provide, as is indicated at 310′, spatial beams aswell as delay-domain combining coefficients and delays (explicitfeedback) or a single or multiple delay identifier (implicit feedback)for each layer and transmit antenna at the base station 200 so as toachieve a predefined property for a communication over the radiochannel. For example, at 308, the complex coefficients and the delays ofthe space-delay precoder may be optimized at the UE 300 to achieve apredefined property for a communication over the radio channel, e.g., bymaximizing a cost function such as mutual information or rate based onlong- and short-term channel state information, as is described in moredetail below. The optimized precoder coefficients and delays are fedback to the gNB 200 over the feedback channel 500 via implicit orexplicit feedback schemes or a combination of both. For example, thefeedback may use CSI indicating CQI, RI, PMI or beam based feedback,delay-domain complex combining coefficients with an explicit feedback ofdelays or an implicit feedback of delays using delay identifiers (DI).

1^(st) Embodiments: Time-Domain Representation of the Space-DelayPrecoder

In accordance with embodiments, the space-delay precoders at 308 modelor define for the associated transmit antenna a plurality of cyclicfilters delaying and weighting a signal to be transmitted with thecorresponding precoder delays and complex precoder coefficients,respectively. Thus, a parametric space-delay precoder scheme is providedwhere the precoder coefficients for the transmit antenna n and rank-rare defined byk _(n,r) =k _(n,r)(1)·δ(t−τ _(n,r)(1))+ . . . +k _(n,r)(l)·δ(t−τ_(n,r)(l))+ . . . +k _(n,r)(L)·δ(t−τ _(n,r)(L))  (1)where k_(n,r)(l) denotes the complex coefficient at delay τ_(n,r)(l).

The delays τ_(n,r)(l), ∀l may be antenna-specific or not. Further, thedelays may be defined for a specific sampling grid such that τ_(n,r)(l)∈

⁺, l=1, 2, . . . , L, where

⁺ denotes the positive integers, or the delays may be defined off thesampling grid, such that τ_(n,r)(l)∈

⁺, l=1, 2, . . . , L, where

⁺ denotes the positive real numbers. The sampling grid is a set ofinteger values of delays for which the channel coefficients areavailable. For the delays defined off the sampling grid, the channelcoefficients are obtained by interpolation. The delays τ_(n,r)(l) may beantenna-specific and layer-specific so that the l-th delay τ_(n,r)(l) ofthe n-th transmit antenna, r-th layer, is different to the l-th delayτ^(k,p)(l) of the k-th transmit antenna, p-th layer,τ_(n,r)(l)≠τ_(k,r)(l),∀n,k,l,r,n≠k,τ_(n,r)(l)≠τ_(n,p)(l),∀n,l,r,p,r≠p, orthe delays τ_(n,r) (l) may be non-antenna-specific andnon-layer-specific so that the l-th delay τ_(n,r) (l) of the n-thtransmit antenna, r-th layer, is identical to the l-th delay τ_(k,p) (l)of the k-th transmit antenna, p-th layer,τ_(n,r)(l)=τ_(k,p)(l),∀n,k,l,r,p.

For the on-grid delays a DFT may be used to calculate the frequencyresponse of the space-delay precoder. The off-grid delays denote anon-uniform sampling of the space-delay precoder (see equation (1)) inthe delay domain, and a DFT may not be used to calculate the frequencyresponse of the space-delay precoder. For non-uniform sampling in delay,the discrete frequency response per subcarrier s is calculated using thenon-uniform discrete Fourier transform (NUDFT) given byK _(n,r)(s)=w(s)·k _(n,r)where

${w(s)} = {{\left( \frac{1}{\sqrt{S}} \right)\left\lbrack {e^{\frac{{- j}\; 2\;\pi\; s}{S}{\tau_{n,r}{(1)}}}\mspace{14mu}\ldots\mspace{14mu} e^{\frac{{- j}\; 2\;\pi\; s}{S}{\tau_{n,r}{(l)}}}\mspace{14mu}\ldots\mspace{14mu} e^{\frac{{- j}\; 2\;\pi\; s}{S}{\tau_{n,r}{(L)}}}} \right\rbrack} \in {\mathbb{C}}^{L}}$is the NUDFT vector and k_(n,r)=[k_(n,r)(1) . . . k_(n,r)(l) . . .k_(n,r)(L)]^(T)∈

^(L) and K _(n,r)(s) is the precoder coefficient associated withsubcarrier s and transmit antenna n and layer r. The complexcoefficients k_(n,r)(l), ∀n, l, r and the delays τ_(n,r)(l), ∀l, n, r ofthe space-delay precoder (see equation (1)) may be calculated at the UEand sent to the gNB with very less feedback.

In the embodiment of FIG. 3 or FIG. 4, the base station 200 mayimplement a conventional precoder, like the one described above withreference to FIG. 2, and a cyclic prefix 210 may be added to the signalto be applied to the antennas 202. In case of using a conventionalprecoder at the precoder, the base statio 200, responsive to thefeedback from the UE 200, may calculate the frequency response of thespace-delay precoder as described above and perform precoding in thefrequency domain responsive to the obtained frequency response persubcarrier. In accordance with embodiments, the base station 200 mayimplement the space-delay precoders as described above. In accordancewith embodiments, the base station 200 may operate on the basis of anoversampled DFT codebook, on the basis of a codebook adapted to antennaarray imperfections, as described by Sutharshun Varatharaajan, MarcusGrolmann, Markus Landmann, “Beamforming codebook adaption to antennaarray imperfections,” European patent application 17154486.9 filed onFeb. 2, 2017, which is incorporated herewith by reference, or on thebasis of a codebook adapted to a predefined antenna response of theantenna array, as described by Venkatesh Ramireddy, Marcus Grolmann,Markus Landmann, “Antenna array codebook with beamforming coefficientsadapted to a predefined antenna response of the antenna array,” Europeanpatent application 17154487.7 filed on Feb. 2, 2017, which isincorporated herewith by reference.

As mentioned above, at the user equipment 300, the complex coefficientsand the delays of the space-delay precoder (see equation (1)) may beoptimized to achieve a predefined property for a communication over theradio channel, e.g., by maximizing a cost function such as mutualinformation or the received signal to noise ratio (SNR) based on long-and short-term channel state information. In case the fed back delaysare on the grid, the system model calculates the frequency response bythe DFT matrices. In the case where the delays are not on the grid, theNUDFT may be used to calculate the frequency response per subcarrier.

In the following a rank-1 transmission is considered and theoptimization problem and the feedback schemes are presented for therank-1 transmission. For simplicity, the subscript r is omitted whenreferring to the rank-1 transmission. However, it is noted that thepresent invention is not limited to such embodiments and may also beimplemented in a communication system employing a higher rank or layercommunication, and the extension to a rank-R transmission isstraightforward.

For a rank-1 transmission, the optimization problem that maximizes theaverage mutual information at the UE may be formulated as

$\begin{matrix}{{\max\limits_{\underset{{\tau_{n}{(l)}},{\forall n},l}{k_{n},{\forall n},}}{\frac{1}{S}{\sum\limits_{s = 1}^{S}\;{\log_{2}\left( {{I_{M} + \frac{{\overset{\_}{H}(s)}{\overset{\_}{K}(s)}{\overset{\_}{K}(s)}^{H}{\overset{\_}{H}(s)}^{H}}{\sigma^{2}}}} \right)}}}}{{{s.t}\mspace{11mu}{\sum\limits_{n = 1}^{N}\;{\sum\limits_{s = 1}^{S}\;{{{w(s)} \cdot k_{n}}}^{2}}}} \leq S}{{{\tau_{n}(l)} \in {\mathbb{R}}^{+}},{\forall l},n,}} & (2)\end{matrix}$where k_(n) is a vector of length L containing the precoder complexcoefficients associated with L delays.

Solving the optimization problem in equation (2), results in theprecoder coefficients and delays that maximize the SNR at the UE so thatapart from the complex coefficients feedback, N·L delays are fed back tothe gNB.

For a rank-1 transmission, for the non-antenna specific case, where thedelays are identical over all antennas, the optimization problem thatmaximizes the average mutual information at the UE is

$\begin{matrix}{{\max\limits_{\underset{\tau_{l},{\forall l}}{k_{n},{\forall n},}}{\frac{1}{S}{\sum\limits_{s = 1}^{S}\;{\log_{2}\left( {{I_{M} + \frac{{\overset{\_}{H}(s)}{\overset{\_}{K}(s)}{\overset{\_}{K}(s)}^{H}{\overset{\_}{H}(s)}^{H}}{\sigma^{2}}}} \right)}}}}{{{s.t}\mspace{11mu}{\sum\limits_{n = 1}^{N}\;{\sum\limits_{s = 1}^{S}\;{{{w(s)} \cdot k_{n}}}^{2}}}} \leq S}} & (3)\end{matrix}$where τ_(l)=τ_(n) (l), ∀n and k_(n) is a vector of length L containingthe precoder complex coefficients associated with L delays.

Solving the optimization problem in equation (3), results in theprecoder coefficients and the delays. The space-delay precoder obtainedfrom solving equation (3) results in the feedback of only L delays tothe gNB instead of N·L delays from equation (2).

Embodiments of the feedback schemes for polarimetric andnon-polarimetric cases are now described for a system employing a rank-1or layer-1 communication. In the case of antenna-specific delays,τ₁(l)≠τ_(n)(l)≠τ_(N)(l), ∀l, i.e., the l_(th) delay is different acrossthe transmit antennas. In case of non-antenna specific delays,τ₁(l)=τ_(n)(l)=τ_(N)(l), ∀l, i.e., the l_(th) delay is identical acrossall transmit antennas.

Non-Polarimetric Case

The complex coefficients of the space-delay precoder are fed back usingcodebook or non-codebook based schemes, also the delays are fed backexplicitly or implicitly. The implicit delay feedback is via a delayidentifier (DI). Each DI refers to a specific set of delays, where eachset is a made up of a combination of delays defined in the sampling gridor not. Each DI may refer to a specific set of delays associated withvectors from a codebook, where each set is a made up of a combination ofdelays defined in the sampling grid or not.

The complex coefficients corresponding to the l_(th)-delay position ofall antennas is collected in a vector ask(l)=[k ₁(l)k ₂(l) . . . k _(N)(l)]^(T)∈

^(N)Feedback Scheme 1: Explicit Feedback of Precoder Coefficients and Delays

Using explicit feedback, per delay, N complex coefficients and N delaysassociated with N transmit antennas, respectively, are be fed back tothe gNB 200. Therefore, the total feedback amounts to N·L complexcoefficients and N·L delays.

In the non-antenna specific case, the feedback amounts to N·L complexcoefficients and L delays.

Codebook-Based Space-Delay Precoding

Considering a double stage precoding structure F=F₁F₂ as describedabove, the corresponding delay-domain precoder k(l) of l_(th)-delay maybe written ask(l)=K ₁(l)K ₂(l),where the delay-specific matrix K₁(l) is a block diagonal matrix of sizeN×2U that contains 2U vectors and K₂(l) is acombining/selection/co-phasing vector of size 2U×1 that combines 2Uvectors.

The beamforming vectors in matrix K₁ may be selected either from anoversampled DFT codebook matrix, similar to F₁, or from an arrayresponse matched codebook designed for arbitrary antenna arrayconfigurations as described in the above mentioned European patentapplications 17154486.9 or 17154487.7, which are incorporated herewithby reference.

Feedback Scheme 2: Implicit Feedback for K₁ and K₂

The feedback corresponding to matrix K₁(l) and vector K₂(l) from the UE300 to the gNB 200 is indicated implicitly via PMI1 and PMI2,respectively. The precoder associated with the l_(th) delay position isspecified by PMI1 and PMI2 along with N delays associated with Ntransmit antennas. Therefore, for L delays, the total feedback amountsto L PMI1s+L PMI2s+N·L delays for the antenna specific case, and to LPMI1s+L PMI2s+L delays for the non-antenna specific case.

In accordance with embodiments, the space-delay precoder correspondingto the l_(th) delay may be decomposed ask(l)=K ₁ K ₂(l)where K₁(l) is a wideband precoder matrix which is identical over alldelays K₁(1)=K₁(l)=K₁ (L), ∀l, and K₂ (l) is the delay specificselection/combining/co-phasing vector. The feedback amounts to 1 PMI1+LPMI2s+N·L delays in the antenna specific case, and to 1 PMI1+L PMI2s+Ldelays in the non-antenna specific case.Feedback Scheme 3: Implicit Feedback for K₁ and Explicit Feedback for K₂

The feedback associated with matrix K₁(l) is similar as described infeedback scheme 2. The feedback for the 2U×1 sized vector K₂ (l) may beindicated to the gNB 200 explicitly with 2U complex entries.

The precoder associated with the l_(th) delay position is specified byPMI1 and 2U complex values along with N delays associated with Ntransmit antennas.

For the L delays, in the antenna specific case the total feedbackamounts to L PMI1s+2·L·U complex coefficients+N·L delays, and in thenon-antenna specific case the feedback equals to L PMI1s+2·L·U complexcoefficients+L delays.

In embodiments employing the above described wideband precoder matrixthe feedback amounts to 1 PMI1+2·L·U complex coefficients+N·L delays forthe antenna specific case, and to 1 PMI1+2·L·U complex coefficients+Ldelays for the non-antenna specific case.

For the feedback schemes 1, 2 and 3, the delays may also be fed back tothe gNB implicitly via delay identifiers (DIs). For antenna specificcase, L DI's may be used for indicating the delays, where each DI isdefined for the delays across the antennas. In the non-antenna specificcase, a single DI suffices to indicate the delays to the gNB, and, sincethe delays are identical across antennas, the DI in this case definesthe delays across the precoder taps.

Table 1 below summarizes the feedback for the feedback schemes discussedabove for the non-polarimetric case.

Non-polarimetric case Feedback for Feedback for Feedback for Feedbackwideband Feedback delays delays for K₁(l) K₁ for K₂(l) Implicit ExplicitAntenna Feedback NL complex coefficients L DIs NL specific scheme 1 caseFeedback L PMI1s 1 PMI1 L PMI2s L DIs NL scheme 2 Feedback L PMI1s 1PMI1 LU L DIs NL scheme 3 complex coefficients Non- Feedback NL complexcoefficients 1 DI L Antenna scheme 1 specific Feedback L PMI1s 1 PMI1 LPMI2s 1 DI L case scheme 2 Feedback L PMI1s 1 PMI1 LU 1 DI L scheme 3complex coefficientsPolarimetric CaseFeedback Scheme 1: Explicit Feedback of Precoder Coefficients and Delays

Using explicit feedback, per delay, N complex coefficients and N delaysassociated with N transmit antennas, respectively, are be fed back tothe gNB 200. Therefore, the total feedback amounts to N·L complexcoefficients and N·L delays.

In the non-antenna specific case, the feedback amounts to N·L complexcoefficients and L delays.

Codebook-Based Space-Delay Precoding

Considering a double stage precoding structure F=F₁F₂ as describedabove, the precoder k(l) of l_(th)-delay may be written ask(l)=K ₁(l)K ₂(l),where the delay-specific matrix K₁(l) is a block diagonal matrix of sizeN×2U that contains 2U vectors and K₂(l) is acombining/selection/co-phasing vector of size 2U×1 that combines 2Uvectors.

The beamforming vectors in matrix K₁ may be selected either from anoversampled DFT codebook matrix or the array response matched codebooksdesigned for arbitrary antenna array configurations as described in theabove mentioned European patent applications 17154486.9 or 17154487.7,which are incorporated herewith by reference.

Feedback Scheme 2: Implicit Feedback for K₁ and K₂

The precoder matrix indices for horizontal polarization and verticalpolarizations are indicated by PMI1h and PMI1v, respectively, forprecoder matrix K₁(l). The feedback corresponding to vector K₂ (l) isindicated to the gNB via PMI2. For the l_(th) delay, PMI1h and PMI1vassociated with K₁(l), respectively, and PMI2 associated with K₂(l),along with N delays are fed back from the UE 300 to the gNB 200.

For the antenna specific case the feedback amounts to L PMI1hs+LPMI1vs+L PMI2+N·L delays, and for the non-antenna specific case thefeedback is L PMI1hs+L PMI1vs+L PMI2+L delays.

If K₁ (l) is chosen as a wideband precoder matrix as described above,for the antenna specific case the total feedback is 1 PMI1h+1 PMI1v+LPMI2+N·L delays, and for the non-antenna specific case, the feedback is1 PMI1h+1 PMI1v+L PMI2+L delays.

Feedback Scheme 3: Implicit Feedback for K₁ and Explicit Feedback for K₂

The feedback associated with matrix K₁(l) is similar as described infeedback scheme 2 of the polarimetric case. For the l_(th) delayposition, the precoder matrix index for horizontal polarization (PMI1h)and the precoder matrix index for vertical polarization (PMI1v) forprecoder matrix K₁(l) and 2U complex coefficients for matrix K₂(l) alongwith N delays are fed back from the UE 300 to the gNB 200.

For L delays, the feedback amounts to L PMI1hs+L PMI1vs+2·L·U complexcoefficients+N·L delays for the antenna specific case, and to L PMI1hs+LPMI1vs+2·L·U complex coefficients+L delays for non-antenna specificcase.

If K₁ (l) is chosen as a wideband precoder matrix as described above,for the antenna specific case the feedback is 1 PMI1h+1 PMI1v+2·L·Ucomplex coefficients+N·L delays, whereas for the non-antenna specificcase the total feedback is 1 PMI1h+1 PMI1v+2·L·U complex coefficients+Ldelays.

For the feedback schemes 1, 2 and 3, the delays may also be fed back tothe gNB implicitly via the delay identifier (DI). For antenna specificcase, L DI's may be used for indicating the delays, where each DI isdefined for the delays across the antennas. In the non-antenna specificcase, a single DI suffices to indicate the delays to the gNB, and, sincethe delays are identical across antennas, the DI in this case definesthe delays across the precoder taps.

Table 2 below summarizes the feedback for the feedback schemes discussedabove for the polarimetric case.

Polarimetric case Feedback Feedback Feedback for for delay- for delay-Feedback Feedback specific wideband specific for delays for delays K₁(l)K₁ K₂(l) Implicit Explicit Antenna Feedback scheme 1 NL complexcoefficients L DI's NL specific Feedback scheme 2 L PMI1h's L PMI1h LPMI2s L DI's NL case + + L PMI1v's 1 PMI1v Feedback scheme 3 L PMI1h's 1PMI1h LU complex L DI's NL + + coefficients L PMI1v's 1 PMI1v Non-Feedback scheme 1 NL complex coefficients 1 DI L Antenna Feedback scheme2 L PMI1h's 1 PMI1h L PMI2s 1 DI L specific + + case L PMI1v's 1 PMI1vFeedback scheme 3 L PMI1h's 1 PMI1h LU complex 1 DI L + + coefficients LPMI1v's 1 PMI1v

In accordance with embodiments, the inventive approach may also beemployed for a MISO system. Based on the channel estimates, the delaysthat correspond to L dominant peaks in the time domain channel may beselected or chosen to be the L delays of the precoder, and based on theMRT (maximum ratio transmission) precoder calculated in the time domain,the L dominant peaks may be selected or chosen to be the L delays of theprecoder.

In case delays of the channel are also estimated, the delays thatcorrespond to the first L dominant peaks of the channel may be selectedor chosen to be the L delays of the precoder, and the delays thatcorresponds to the first L dominant peaks of the MRT precoder may beselected or chosen to be the L delays of the precoder.

In case the channel delays are off the grid, a high-resolution parameterestimation approach may be used to estimate the delays, for example thespace alternating generalized expectation-maximization (SAGE) algorithm(see reference [6]).

Some of the embodiments of the present invention have been describedabove with reference to two-dimensional (2D) uniform planar arrays(UPAs) using dual-stage/double-structure codebooks. However, the presentinvention is not limited to such embodiments and may also be implementedusing triple-structure codebooks in accordance with the 5G or NR (NewRadio) standard. Further, the present invention is not limited to 2Darrays. The inventive approach is equally applicable to any arbitraryantenna array configuration, like a one-dimensional (1D) uniform lineararray (ULAs) on a three-dimensional (3D) array antenna, like cylindricalarrays or conical arrays. Three-dimensional (3D) array antennas aredescribed, e.g., in PCT Patent Application PCT/EP2017/064828,“Transmitter, Receiver, Wireless Communication Network and Methods forOperating the Same” filed on 16 Jun. 2017, which is incorporatedherewith by reference.

When considering a multi-panel array with P_(R) panels in each row andP_(C) panels in each columns, the total number of panels is given byP=P _(R) P _(C).

The number of antennas per panel remains the same as discussed above forthe dual stage structure. For such a multi-panel antenna structure, theprecoder is given by a ternary/triple-stage structureF=F ₃ F ₁ F ₂where F₃ is a wideband phase compensation matrix of size P×N, which isused to compensate for the phase offset between multiple panels given byF ₃=[e ^(jθ) ¹ e ^(jθ) ² . . . e ^(jθ) ^(P) ]^(T) ⊗I _(N)where e^(jθ) ^(p) is the phase compensation factor per panel. Here Ndenotes the total number of antennas per panel including allpolarizations/orientations. The matrices F₁ and F₂ are used forprecoding within a panel and have the same functionality as described inthe dual-stage structure.

For the present invention, the precoder coefficients of delay 1 andpanel p may be written ask(l,p)=K ₃(p)K ₁(l,p)K ₂(p).

The matrix K₃ (p) is a wideband matrix defined by the phase compensationfactor given byK ₃(p)=e ^(jθ) ^(p) ⊗I _(N),and the matrix K₁ and vector K₂ may be identical or different across thepanels i.e., they can be panel specific or panel non-specific.

In the panel specific case, feedback for matrix K₁ and vector K₂ alongwith the phase compensation factor per panel, respectively, is fed backto the gNB.

In the panel non-specific case, the feedback for matrix K₁ and vector K₂for a single panel along with the phase compensation factors per panelis fed back to the gNB.

For the panel specific and panel non-specific case, the feedback formatrix K₁ and vector K₂ described in the feedback schemes 1, feedback 2and feedback 3 for the polarimetric and non-polarimetric case applies.

The feedback for the phase compensation factors across panels may beimplicit via the index (PMI3) chosen or selected from a modulationscheme constellation or from a DFT codebook or may be explicit. For theexplicit case, P phase compensation factors are fedback, whereas in theimplicit case, PMI3 is used for the feedback.

Table 3 below summarizes the feedback for matrix K₃ in the panelspecific and panel non-specific cases.

Total feedback for ternary/triple Feedback scheme precoder structurePanel Explicit feedback P angles + Feedback of K₁ and K₂ per specific ofK₃ and panel case feedback of K₁ and K₂ Implicit feedback 1 PMI3 +Feedback of Feedback of of K₃ and K₁ and K₂ per panel feedback of K₁ andK₂ Panel Explicit feedback P angles + Feedback of K₁ and K₂ for non- ofK₃ and single panel specific feedback of K₁ and K₂ Implicit feedback 1PMI3 + Feedback of K₁ and K₂ for of K₃ and single panel feedback of K₁and K₂

2^(nd) Embodiments: Frequency-Domain Representation of the Space-DelayPrecoder

In the embodiments described so far the space-delay precoder k(l) isrepresented in the time domain. However, the inventive approach is notlimited to such embodiments, and in accordance with further embodimentsof the inventive approach the space-delay precoder k(l) is representedin the frequency domain.

The feedback schemes, which are based on a frequency-domainrepresentation of the space-delay precoder, are now described fornon-polarimetric cases in a system employing a rank-1 or layer-1communication. In the case of antenna-specific delays,τ₁(l)≠τ_(n)(l)≠τ_(N)(l), ∀l, i.e., the l_(th) delay is different acrossthe transmit antennas. In case of non-antenna specific delays,τ₁(l)=τ_(n)(l)=τ_(N) (l), ∀l, i.e., the l_(th) delay is identical acrossall transmit antennas. As mentioned above, the present invention is notlimited to rank-1 embodiments and may also be implemented in acommunication system employing a higher rank or layer communication, andthe extension to a rank-R transmission is straightforward. Further, theextension to polarimetric cases is straightforward (see above).

The complex coefficients describing the space-delay precoder may be fedback using codebook and non-codebook based schemes, e.g., in a way asdescribed above with reference to the first embodiment, and the delaysmay be fed back explicitly or implicitly. The implicit delay feedbackmay use a delay identifier, DI. Each DI may include indices associatedwith respective column vectors of a codebook matrix used at thetransmitter,

The space-delay precoder k(l) is described using the complexcoefficients corresponding to the l_(th)-delay position of all antennasas followsk(l)=[k ₁(l)k ₂(l) . . . k _(N)(l)]^(T)∈

^(N)

The space-delay precoder k(l) may be transformed to the frequency-domainby applying a NU-DFT matrix. To do this, the vectors k(l) for the Ldelays are stacked in a matrix {tilde over (K)},{tilde over (K)}=[k(1) . . . k(l) . . . k(L)]∈

^(N×L).

In the following, the antenna-specific and the antenna-non-specificcases are treated separately. Further, in the following, the doublestage precoder structure used in 3GPP (see reference [7]) is adopted anda rank-1 transmission is considered. Moreover, in the following weconsider the case of dual-polarized antenna arrays, such that P=2. Thenthe precoder for a subcarrier s is given by

${{F(s)} = {{F_{1}{F_{2}(s)}} = {\sum\limits_{u = 1}^{2\; U}{{\overset{\_}{s}}_{u}{f_{2,u}(s)}}}}},{where}$${{\overset{\_}{s}}_{u} = {\begin{bmatrix}s_{u} \\0_{\frac{N}{2}}\end{bmatrix} \in {\mathbb{C}}^{N}}},{{\forall u} = {1\mspace{14mu}\ldots\mspace{14mu} U}},{{\overset{\_}{s}}_{U + u} = {\begin{bmatrix}0_{\frac{N}{2}} \\s_{u}\end{bmatrix} \in {\mathbb{C}}^{N}}},{{\forall u} = {1\mspace{14mu}\ldots\mspace{14mu} U}},{and}$f_(2,u)(s)∈

denotes the complex coefficient associated with beam u and subcarrier s.

Collecting the precoders for all subcarriers in the matrix F, oneobtainsF=F ₁[F ₂(0)F ₂(2) . . . F ₂(S−1)]=F ₁ F ₂(a) Antenna-Specific Case:

For the antenna-specific case, the corresponding frequency-domainprecoder for {tilde over (K)} is given byF={hacek over (K)}{hacek over (W)},where the entries of {tilde over (K)} are arranged in a block-diagonalmatrix {hacek over (K)},

$\overset{\Cup}{K} = {\begin{bmatrix}k_{1}^{T} & 0_{L}^{T} & \; & 0_{L}^{T} \\0_{L}^{T} & k_{2}^{T} & \; & 0_{L}^{T} \\\vdots & \vdots & \ldots & \vdots \\0_{L}^{T} & 0_{L}^{T} & \; & k_{N}^{T}\end{bmatrix} \in {\mathbb{C}}^{N \times L\; N}}$with k_(n)=[k_(n)(1) . . . k_(n)(l) . . . k_(n)(L)]^(T)∈

^(L×1) being the delay-domain precoder coefficients for the space-delayprecoder for the L delays and the n-th transmit antenna, and 0_(L) isthe all zero-element column vector of size L. The NU-DFT matrix {hacekover (W)} of size LN×S is given by{hacek over (W)}=[{hacek over (W)} ₁ {hacek over (W)} ₂ . . . {hacekover (W)} _(N)]^(T),where the NU-DFT submatrix {hacek over (W)}_(n)=[w_(n,1) w_(n,2) . . .w_(n,L)]∈

^(S×L) contains L vectors

$w_{n,l} = {\left\lbrack {1\mspace{14mu}\ldots\mspace{14mu} e^{\frac{{- j}\; 2\;\pi\; s}{S}\tau_{n{(l)}}}\mspace{14mu}\ldots\mspace{14mu} e^{\frac{{- j}\; 2\;{\pi{({S - 1})}}}{S}{\tau_{n}{(l)}}}}\; \right\rbrack^{T} \in {{\mathbb{C}}^{S \times 1}.}}$

The vector w_(n,l) depends on the delay τ_(n)(l) and antenna index n.

The number of delays defined per antenna can be different.

(b) Non-Antenna-Specific Case:

For the non-antenna-specific case, the corresponding frequency-domainprecoder for {tilde over (K)} is given byF={tilde over (K)}{tilde over (W)},where {tilde over (W)}=[w₁ w₂ . . . w_(L)]^(T)∈

^(L×S) is the NU-DFT matrix defined for L delays with w_(l) being theNU-DFT vector associated with delay τ(l),

$w_{l} = {\left\lbrack {1\mspace{14mu}\ldots\mspace{14mu} e^{\frac{{- j}\; 2\;\pi\; s}{S}{\tau{(l)}}}\mspace{14mu}\ldots\mspace{14mu} e^{\frac{{- j}\; 2\;{\pi{({S - 1})}}}{S}{\tau{(l)}}}}\; \right\rbrack^{T} \in {{\mathbb{C}}^{S \times 1}.}}$

Implicit Delay (DI) Feedback

In accordance with embodiments, the delay of the space-delay precoderk(l) represented in the frequency domain may be fed back implicitly,e.g., using one or more indices associated with respective columnvectors of a frequency-domain codebook matrix used at the receiver. Forexample, a precoding matrix identifier (PMI) may be employed, and thePMI may correspond to a set of indices, where each index refers to aspecific column in a DFT codebook. In accordance with embodiments, afirst number of indices in the PMI indicates the respective beams, asecond number of indices in the PMI indicates the respectivedelay-domain precoder coefficients, and a third number of indices, whichare the indices of the delay identifier, DI.

(a) Codebook-Based DI Feedback

In the case of an implicit DI feedback, in accordance with embodiments,the DI contains a set of L indices which are associated with columnvectors of a frequency-domain codebook matrix D. The delays τ(l)∈

, ∀l are discretized and are given by elements of a set

=[0, . . . , SO_(f)−1]. Moreover, each value in

is associated with a column vector of the frequency-domain codebookmatrix D. Therefore, the NU-DFT vectors w_(l), ∀l may be represented byDFT-vectors as follows:

${d_{i} = {\left\lbrack {1\mspace{14mu}\ldots\mspace{14mu} e^{\frac{{- j}\; 2\;\pi\; i}{O_{f}S}}\mspace{14mu}\ldots\mspace{14mu} e^{\frac{{- j}\; 2\;\pi\;{i{({S - 1})}}}{O_{f}S}}}\; \right\rbrack^{T} \in {\mathbb{C}}^{S \times 1}}},$i∈

with O_(f) being the oversampling factor of the codebook DFT-matrixD=[d₀, d₁, . . . , d_(SO) _(f) ⁻¹] and j=√{square root over (−1)}.

The codebook matrix D is parameterized by the number of subcarriers andthe oversampling factor O_(f).

When O_(f)=1, the codebook matrix D is given by an S×S DFT-matrix.

When O_(f)>1, the codebook matrix D is given by an oversampledDFT-matrix of size S×(O_(f)S−1).

The oversampling factor O_(f) is signaled from the transmitter to thereceiver such that the receiver may construct the codebook matrix.

Based on the above definition of the frequency-domain codebook matrix D,the corresponding frequency-domain precoder for {tilde over (K)} isdefined byF={tilde over (K)}[w ₁ w ₂ . . . w _(L)]^(T) with w _(l) ∈D,∀l.(b) Double-Stage Precoding Structure F=F₁F₂—Identical Delays for all 2UBeams

In accordance with embodiments, similar to the frequency-domaindouble-stage precoder structure F=F₁F₂, the space-delay precoder for thel-th delay may be expressed ask(l)=K ₁(l)K ₂(l)where K₁ is a matrix of size N×2U that contains 2U spatial beams, andK₂(l) is a vector of size 2U×1,K ₂(l)=[K _(2,1)(l) . . . K _(2,u)(l) . . . K _(2,2U)(l)]^(T)∈

^(2U×1)with K_(2,u)(l) being a scalar complex delay-domain combiningcoefficient associated with u-th beam and the l-th delay. When K₁ (l) isa wideband matrix, the space-delay precoder matrix {tilde over (K)} maybe expressed as{tilde over (K)}=K ₁ K ₂where K₁ is identical to matrix F₁, and K₂=[K₂(1) . . . K₂(l) . . .K₂(L)]∈

^(2U×L). Therefore, the double-stage precoding structure F=F₁F₂ may bewritten asF=K ₁ K ₂ {tilde over (W)},F ₁ =K ₁ ,F ₂ =K ₂ {tilde over (W)}.

The delays τ(l), ∀l in the DI used in matrix {tilde over (W)} areidentical to all 2U beams in matrix K₁.

(c) Extension to Beam-Specific Delays—Polarization and Beam DependentDelays

In accordance with embodiments, when K₁(l) is a wideband matrix and thecombination of beams for the l-th delay may differ to other delays, andthe delays associated with the 2U beams may be different. Therefore the2U beams may be associated with 2U DIs. The u-th DI is then associatedwith the beam index u and with L delays τ_(u)(l), l=1, . . . L, wherethe L delays may be identical or non-identical for different beams.Also, each beam can have different number of delays L. The frequencydomain precoder may then be represented byF=K ₁ ·{hacek over (K)} ₂ ·W,where the matrix {hacek over (K)}₂ is the space-delay-domain combiningcoefficient matrix, defined as

$\overset{\Cup}{K} = {\begin{bmatrix}K_{2,1} & 0_{\overset{\_}{L}}^{T} & \; & 0_{\overset{\_}{L}}^{T} \\0_{\overset{\_}{L}}^{T} & K_{2,2} & \; & 0_{\overset{\_}{L}}^{T} \\\vdots & \vdots & \ldots & \vdots \\0_{\overset{\_}{L}}^{T} & 0_{\overset{\_}{L}}^{T} & \; & K_{2,{2\; U}}\end{bmatrix} \in {\mathbb{C}}^{2\; U \times 2\;\overset{\_}{L}\; U}}$with K_(2,u)=[K_(2,u)(1) . . . K_(2,u)(l) . . . K_(2,u)(L)]∈

^(1×L) being the delay-domain combining coefficients associated withbeam u. Furthermore, W is given byW=[W ₁ W ₂ . . . W _(2U)]^(T)∈

^(2LU×S)with W_(u)=[w_(u,1) w_(u,2) . . . w_(u,L) ]∈

^(S×L) being the DFT matrix associated with beam u, whose L columns areselected from the codebook D.

The matrix F₂ containing the frequency-domain combining-coefficientsf_(2,u) may be expressed asF ₂=[f _(2,1) f _(2,2) . . . f _(2,2U)]^(T)where

$f_{2,u} = {{{\sum\limits_{l = 1}^{\overset{\_}{L}}\;{w_{u,l}{K_{2,u}(l)}\mspace{14mu}{with}\mspace{14mu} w_{u,l}}} \in {D.}} = {W_{u}{K_{2,u}^{T}.}}}$

Therefore, the precoder F may then be written as

$F = {\sum\limits_{u = 1}^{2\; U}\;{{\overset{\_}{s}}_{u} \cdot \left( {\sum\limits_{l = 1}^{\overset{\_}{L}}\;{w_{u,l}^{T}{K_{2,u}(l)}}} \right)}}$(c.1) Beam-Specific Delays—Special Case of Polarization-Independent andBeam-Dependent Delays

In accordance with embodiments, the delays τ_(u)(l) arepolarization-independent and beam-dependent, and the following applies:τ_(u)(l)=τ_(U+u)(l),l=1, . . . ,L,∀u.

Then, the following relation holds for the frequency domain vectorw_(u,1):w _(u,l) =w _(U+u,l) ,∀l,∀u.

Therefore, instead of the 2U DI feedback only U DIs need to be fed backto the transmitter.

(c.2) Beam-Specific Delays—Special Case of Polarization-Dependent andBeam-Independent Delays

In accordance with embodiments, the delays are polarization-dependentand beam-independent, and the following applies:τ_(u)(l)=τ⁽¹⁾(l), τ_(U+u)(l)=τ⁽²⁾(l),∀l,u=1, . . . ,U,where τ⁽¹⁾(l)≠τ⁽²⁾(l).

Then, the following relation holds for the frequency domain vectorw_(u,l)w _(u,l) =w _(l) ⁽¹⁾ ,w _(U+u,l) =w _(l) ⁽²⁾ ∀l,u=1, . . . ,Uwith w_(l) ⁽¹⁾≠w_(l) ⁽²⁾.

Therefore, instead of the 2U DI feedback only two DIs, 2 DIs, need to befed back to the transmitter, where the first DI refers to the delays ofthe first polarization of the antenna array, and the second DI refers tothe delays of the second polarization of the antenna array

The following table summarizes the total amount of feedback for matrixK₂ and the number of DIs for various feedback embodiments.

Delay-identifier: Number of indices to indicate the Number ofdelay-domain columns of the complex delay-domain frequency domaincombining coefficients (K₂) codebook Identical delays for 2LU complexcoefficients 1 DI all 2U beams → see (b) Polarization and beam 2LUcomplex coefficients 2U DIs dependent delays → see (c) Polarization 2LUcomplex coefficients U DIs independent and beam dependent delays → see(c.1) Polarization dependent 2LU complex coefficients 2 DIs and beamindependent delays → see (c.2)(c.3) Beam-Specific Delays—Special Case of d Identical Delays Out of LDelays

In accordance with embodiments, d indices out of L indices in a DIassociated with the u-th beam may be identical to the delay indices ofthe DIs associated with other beams. In such a case, the DI of the u-thbeam may have only L-d indices instead of L indices.

In addition to beam-specific DIs that contain indices for specificspatial beams, a DI common to X (X=1 . . . PU) spatial beams may be usedto denote indices common to X spatial beams. Such multiple common DIsmay become relevant when there are multiple sets of identical delaysamong DIs of different spatial beams.

The DI configuration may be signaled from the transmitter to thereceiver. A DI configuration for example may contain information about:

-   -   total number of indices per beam-specific DI, or    -   number of common DIs, number of indices per common DI.        (c.4) Beam-Specific Delays—Restriction of Delays

In accordance with further embodiments, for each beam the L delays maybe centered or restricted to lie around a single mean delay. Then, thefrequency domain codebook matrix W _(u) for the u-th beam is given by

${{\hat{W}}_{b_{u,1}} = {\left\lbrack {d_{({b_{u,1} - \frac{c}{2}})}\mspace{14mu}\ldots\mspace{14mu} d_{b_{u,1}}\mspace{14mu}\ldots\mspace{14mu} d_{({b_{u,1} + \frac{c}{2}})}} \right\rbrack \in {\mathbb{C}}^{S \times \overset{\_}{L}}}},$where L=C+1 with C being a window parameter, and b_(u,1) is the indexassociated with the mean delay, see FIG. 5 which illustrates the L delayindices for the u-th beam centered around the mean delay index b_(u,1).The window-size parameter C can be identical or different for thespatial beams, and is signaled via a control channel or via higherlayer-signaling from the transmitter to the receiver.

For each beam, L delay-domain complex combing-coefficients coefficientsare fed back to the transmitter. However, instead of the feedback of Ldelays per beam, a single index b_(u,1) of the associated mean delay issufficient to be fed back to the transmitter.

For example, when the window-size parameter C is identical for allbeams, the total feedback amounts to 2LU complex delay-domain combiningcoefficients and 2U DIs for 2U beams, where each DI contains only asingle index.

The optimized mean delay may lie at the beginning or at the end of thedefined sampling grid as shown in FIG. 6. FIG. 6(a) and FIG. 6(b)illustrate possible locations for the mean delay of FIG. 5 lying at thebeginning and/or at the end of the sampling grid. In such cases, amodulo operation may be used to calculate the correct positions(indices) of the delays around the mean delay. The indices for which themodulo operation is needed are highlighted in the boxes b1, b2.

Extension to Multiple Mean Delays Per Beam:

Instead of having a single mean delay, in accordance with embodiments,the above case may be extended to multiple mean delays. Similar to thesingle mean delay case, C delays are optimized around each mean delay asshown in FIG. 7, which illustrates the C delay indices centered aroundtwo mean delay indices b_(u,1) and b_(u,2) for the u-th beam.

For example, when the window-size parameter C is identical for all beamsand all mean delays, for {tilde over (L)} mean delays per beam, thetotal feedback amounts to 2L{tilde over (L)}U complex delay-domaincombining coefficients and 2U DIs for 2U beams, where each DI contains{tilde over (L)} indices.

(c.5) Beam-Specific Delays—Kronecker Product Structure for Delay-DomainCombining Coefficients for the Case of Restricted Delays

In accordance with yet other embodiments, L complex delay-domaincoefficients of the u-th beam associated with the mean delay indexb_(u,{circumflex over (l)}) are used to calculate thecombining-coefficients of all other 2U−1 beams. In the following, weconsider a single mean delay and a single spatial beam. Collecting Ldelay-combining coefficients associated with the u-th beam and meandelay index b_(u,{circumflex over (l)}) ({circumflex over (l)} rangesfrom 1 to 2U) in a row vector K_(2,u)∈

^(1×L) , the complex delay-domain combining-coefficients of theremaining 2U−1 beams (g≠u) associated with the mean delay indexb_(u,{circumflex over (l)}) can be calculated by{circumflex over (K)} ₂=[e _(1,u) . . . e _(g,u) . . . e _(2U−1,u)]^(T)⊗K _(2,u)∈

^(2U×L)where e_(g,u) is the scalar complex coefficient associated with the g-thbeam. FIG. 8 illustrates the calculating of the complex coefficients ofthe (2U−1) beams with respect to the reference beam (box R) for the meandelay b_(u,{circumflex over (l)})

Note that for the above Kronecker product structure, in addition to thefeedback of the 2U delay-domain combining-coefficient vectors K_(2,u),the complex-combing coefficients e_(g,u) need to be fedback to thetransmitter.

Normalization of Delays

In accordance with other embodiments the delays may be normalized withrespect to a single reference delay. A reference delay may be set andthe L delays corresponding to all beams or all antennas are subtractedfrom a single reference delay. Any l-th delay in the set of L delays maybe chosen as the reference delay. In the case of explicit feedback ofdelays, the L−1 delay differences are feed back to the transmitterinstead of the delays. In the case of implicit feedback of delays, L−1delay differences are given by elements of the set

=[0, . . . , SO_(f)−1], and the DIs contain indices associated with thedelay differences.

Specific Case of Per Beam/Antenna Normalization:

A reference delay may also be set per beam or per antenna and the Ldelays corresponding to each beam or each antenna are subtracted fromthe beam- or antenna-specific reference delay. In the case of implicitfeedback of delays, the L−1 delay differences are given by elements ofthe set

=[0, . . . , SO_(f)−1], and the DIs contain indices associated with thedelay differences.

In the embodiments described herein the feedback may be signaled using afeedback channel between a user equipment and a base station as shown inFIG. 2, FIG. 3 or FIG. 4. The feedback may also be signaled ortransmitted via a control channel, like the PUCCH, or it may signaledvia higher layer signaling, like RRC signaling.

The embodiments of the present invention have been described above withreference to a communication system employing a rank-1 or layer-1communication. However, the present invention is not limited to suchembodiments and may also be implemented in a communication systememploying a higher rank or layer communication. In such embodiments, thefeedback includes the delays per layer and the complex precodercoefficients per layer.

The embodiments of the present invention have been described above withreference to a communication system in which the transmitter is a basestation serving a user equipment, and the receiver is the user equipmentserved by the base station. However, the present invention is notlimited to such embodiments and may also be implemented in acommunication system in which the transmitter is a user equipment servedby a base station, and the receiver is the base station serving the userequipment.

Although some aspects of the described concept have been described inthe context of an apparatus, it is clear that these aspects alsorepresent a description of the corresponding method, where a block or adevice corresponds to a method step or a feature of a method step.Analogously, aspects described in the context of a method step alsorepresent a description of a corresponding block or item or feature of acorresponding apparatus.

Various elements and features of the present invention may beimplemented in hardware using analog and/or digital circuits, insoftware, through the execution of instructions by one or more generalpurpose or special-purpose processors, or as a combination of hardwareand software. For example, embodiments of the present invention may beimplemented in the environment of a computer system or anotherprocessing system. FIG. 9 illustrates an example of a computer system700. The units or modules as well as the steps of the methods performedby these units may execute on one or more computer systems 700. Thecomputer system 700 includes one or more processors 702, like a specialpurpose or a general purpose digital signal processor. The processor 702is connected to a communication infrastructure 704, like a bus or anetwork. The computer system 700 includes a main memory 706, e.g., arandom access memory (RAM), and a secondary memory 708, e.g., a harddisk drive and/or a removable storage drive. The secondary memory 708may allow computer programs or other instructions to be loaded into thecomputer system 700. The computer system 700 may further include acommunications interface 710 to allow software and data to betransferred between computer system 700 and external devices. Thecommunication may be in the form electronic, electromagnetic, optical,or other signals capable of being handled by a communications interface.The communication may use a wire or a cable, fiber optics, a phone line,a cellular phone link, an RF link and other communications channels 712.

The terms “computer program medium” and “computer readable medium” areused to generally refer to tangible storage media such as removablestorage units or a hard disk installed in a hard disk drive. Thesecomputer program products are means for providing software to thecomputer system 700. The computer programs, also referred to as computercontrol logic, are stored in main memory 706 and/or secondary memory708. Computer programs may also be received via the communicationsinterface 710. The computer program, when executed, enable the computersystem 700 to implement the present invention. In particular, thecomputer program, when executed, enable processor 702 to implement theprocesses of the present invention, such as any of the methods describedherein. Accordingly, such a computer program may represent a controllerof the computer system 700. Where the disclosure is implemented usingsoftware, the software may be stored in a computer program product andloaded into computer system 700 using a removable storage drive, aninterface, like communications interface 710.

The implementation in hardware or in software may be performed using adigital storage medium, for example cloud storage, a floppy disk, a DVD,a Blue-Ray, a CD, a ROM, a PROM, an EPROM, an EEPROM or a FLASH memory,having electronically readable control signals stored thereon, whichcooperate (or are capable of cooperating) with a programmable computersystem such that the respective method is performed. Therefore, thedigital storage medium may be computer readable.

Some embodiments according to the invention comprise a data carrierhaving electronically readable control signals, which are capable ofcooperating with a programmable computer system, such that one of themethods described herein is performed.

Generally, embodiments of the present invention may be implemented as acomputer program product with a program code, the program code beingoperative for performing one of the methods when the computer programproduct runs on a computer. The program code may for example be storedon a machine readable carrier.

Other embodiments comprise the computer program for performing one ofthe methods described herein, stored on a machine readable carrier. Inother words, an embodiment of the inventive method is, therefore, acomputer program having a program code for performing one of the methodsdescribed herein, when the computer program runs on a computer.

A further embodiment of the inventive methods is, therefore, a datacarrier (or a digital storage medium, or a computer-readable medium)comprising, recorded thereon, the computer program for performing one ofthe methods described herein. A further embodiment of the inventivemethod is, therefore, a data stream or a sequence of signalsrepresenting the computer program for performing one of the methodsdescribed herein. The data stream or the sequence of signals may forexample be configured to be transferred via a data communicationconnection, for example via the Internet. A further embodiment comprisesa processing means, for example a computer, or a programmable logicdevice, configured to or adapted to perform one of the methods describedherein. A further embodiment comprises a computer having installedthereon the computer program for performing one of the methods describedherein.

In some embodiments, a programmable logic device (for example a fieldprogrammable gate array) may be used to perform some or all of thefunctionalities of the methods described herein. In some embodiments, afield programmable gate array may cooperate with a microprocessor inorder to perform one of the methods described herein. Generally, themethods are advantageously performed by any hardware apparatus.

While this invention has been described in terms of several embodiments,there are alterations, permutations, and equivalents which fall withinthe scope of this invention. It should also be noted that there are manyalternative ways of implementing the methods and compositions of thepresent invention. It is therefore intended that the following appendedclaims be interpreted as including all such alterations, permutationsand equivalents as fall within the true spirit and scope of the presentinvention.

REFERENCES

-   [1] Erik Dahlman, Stefan Parkvall, Johan Sköld, “4G:    LTE/LTE-Advanced for Mobile Broadband,” Academic Press, 2011.    (ISBN:012385489X 9780123854896)-   [2] 3GPP TR 36.897 V13.0.0, “3rd Generation Partnership Project;    Technical Specification Group Radio Access Network; Study on    elevation beamforming/Full-Dimension (FD) Multiple Input Multiple    Output (MIMO) for LTE (Release 13),” June 2015.-   [3] Cheng et al., “Two-dimensional Discrete Fourier Transform based    Codebook for Elevation Beamforming,” United States Patent    Application, US 2016/0173180 A1, June 2016.-   [4] 3GPP TS 36.211, “Technical Specification Group Radio Access    Network; Evolved Universal Terrestrial Radio Access (E-UTRA);    Physical Channels and Modulation (Release 10),” V10.4.0, December    2011.-   [5] 3GPP TR 38.802 V14.1.0, “3rd Generation Partnership Project;    Technical Specification Group Radio Access Network; Study on New    Radio access technology: Physical layer aspects (release 14),” June    2017.-   [6] J. A. Fessler and A. O. Hero, “Space-alternating generalized    expectation-maximization algorithm,” IEEE transactions on Signal    Processing, vol. 42, no. 10, pp. 2664-2677, October 1999.-   [7] 3GPP TS 38.214 V13.0.0, “3rd Generation Partnership Project;    Technical Specification Group Radio Access Network; NR; Physical    layer procedures for data (Release 15),” January 2018.

The invention claimed is:
 1. A receiver, configured to receive andprocess a radio signal received via a radio channel from a transmitteremploying a plurality of antennas or antenna ports, determine, based onthe received signal, complex precoder coefficients and delays ofrespective space-delay precoders for each layer and transmit beam at thetransmitter so as to achieve a predefined property for a communicationover the radio channel, each space-delay precoder comprising: adouble-stage precoding structure, the double-stage precoding structureincluding a beamforming matrix that contains PU vectors to form PUspatial beams, U=total number of vectors per polarization, and P=numberof polarizations, where P=1 for co-polarized antenna arrays at thetransmitter and P=2 dual-polarized antenna arrays at the transmitter,wherein the double-stage precoding structure further includes aspace-delay-domain combining coefficient vector or matrix includingcomplex delay-domain combining-coefficients associated with the beamsand the delays, provide a feedback to the transmitter the determineddelays and the determined complex precoder coefficients, the complexprecoder coefficients including the complex delay-domaincombining-coefficients, wherein the feedback includes a precoding matrixidentifier (PMI), the PMI indicating a number of indices of the vectorsassociated with the spatial beams, respective complex delay-domaincombining-coefficients, and a number of indices of the delays associatedwith respective column vectors of a codebook matrix.
 2. The receiver ofclaim 1, wherein the respective column vectors are non-uniform discreteFourier transform (NU-DFT)-based vectors.
 3. The receiver of claim 2,wherein the space-delay precoder for each layer and S subbands is givenby F=K₁K₂{tilde over (W)}, wherein K₁ is the beamforming matrix, K₂ isthe space-delay-domain combining coefficient vector or matrix includingthe complex delay-domain combining-coefficients, and {tilde over (W)} isa matrix of respective NU-DFT-based vectors associated with the delays.4. The receiver of claim 1, wherein the U vectors of the beamformingmatrix are identical for both polarizations, and the PMI indicates Uindices of the vectors associated with the spatial beams.
 5. Thereceiver of claim 1, wherein the U vectors of the beamforming matrix areselected from a DFT-based codebook matrix.
 6. The receiver of claim 1,wherein the codebook matrix includes a DFT-based matrix.
 7. The receiverof claim 1, wherein the delays, τ(l)∈

, ∀l, are discretized and are given by elements of a set

=[0, . . . , SO_(f)−1], and each value in

is associated to a column vector of a frequency-domain codebook matrixD, with l=1, . . . , L, S=total number of subbands.
 8. The receiver ofclaim 1, wherein a delay identifier (DI), is associated with a spatialbeam, and the feedback includes PU DIs for PU spatial beams, U=total ofvectors per polarization, P=number of polarizations, where P=1 forco-polarized antenna arrays at the transmitter and P=2 dual-polarizedantenna arrays at the transmitter.
 9. The receiver of claim 1, wherein(i) in case of identical delays for all PEI beams, the feedback includesone delay identifier, 1 DI, for the PU beams, or (ii) in case ofpolarization-dependent and beam-dependent delays, the feedback includesPU delay identifiers, PU DIs, for the PU beams, each DI indicatingindices of delays associated with a single spatial beam, or (iii) incase of polarization-independent and beam-dependent delays, the feedbackincludes U delay identifiers, U DIs, for the FU beams, or (iv) in caseof polarization-dependent and beam-independent delays, the feedbackincludes two delay identifiers, P DIs, for the PU beams.
 10. Thereceiver of claim 8, wherein the number of indicated indices in the DIsis identical or different with respect to the spatial beams.
 11. Thereceiver of claim 1, wherein d delay indices out of L delay indices in adelay identifier, DI, associated with a u-th spatial beam, are identicalto the delay indices of DIs associated with one or more other spatialbeams, then the DI of the u-th spatial beam contains L-d indices insteadof L indices.
 12. The receiver of claim 1, wherein the feedbackincludes, in addition to beam-specific DIs a common DI, the common DIindicating indices common to one or more spatial beams.
 13. The receiverof claim 1, wherein a DI configuration may be signaled from thetransmitter to the receiver, wherein the DI configuration may containinformation about: total number of indices for a beam-specific DI,number of common DIs, and/or number of indices per common DI.
 14. Thereceiver of claim 1, wherein, in case the delays associated with aspatial beam are within a predefined window around a single mean delay,a delay identifier for the spatial beam includes only a single indexassociated with the mean delay.
 15. The receiver of claim 13, wherein,in case of PU beams, the feedback includes PU DIs for the PU beams, witheach DI indicating only a single index.
 16. The receiver of claim 13,wherein the feedback includes a single or multiple DIs for the spatialbeams, with each DI indicating a single or multiple indices, and eachindex is associated with a specific mean delay of the beam.
 17. Thereceiver of claim 13, wherein the PU spatial beams have the same ordifferent mean delays.
 18. The receiver of claim 1, wherein the receiveris configured to feed back the delays of the space-delay precoderimplicitly by setting a reference delay to all beams, L−1 indicesassociated with the L−1 delay differences with respect to the referencedelay are fed back.
 19. A transmitter, comprising: an antenna arrayhaving a plurality of antennas for a wireless communication with one ormore receivers; and a precoder connected to the antenna array, theprecoder to apply a set of beamforming weights to one or more antennasof the antenna array to form, by the antenna array, one or more transmitbeams, wherein the transmitter is configured to determine thebeamforming weights responsive to a feedback received from a receiver,the feedback indicating delays and complex precoder coefficients, theindicated delays and complex precoder coefficients obtained based onrespective space-delay precoders for each layer and transmit beam at thetransmitter so as to achieve a predefined property for a communicationover a radio channel to the receiver, the complex precoder coefficientsincluding complex delay-domain combining-coefficients, and eachspace-delay precoder comprising: a double-stage precoding structure, thedouble-stage precoding structure including a beamforming matrix thatcontains PU vectors to form PU spatial beams for the antennas at thetransmitter, U=total number of vectors per polarization, and P=number ofpolarizations, where P=1 for co-polarized antenna arrays at thetransmitter and P=2 dual-polarized antenna arrays at the transmitter,wherein the double-stage precoding structure further includes aspace-delay-domain combining coefficient vector or matrix including thecomplex delay-domain combining-coefficients associated with the beamsand the delays, and wherein the feedback includes a precoding matrixidentifier (PMI), the PMI indicating a number of indices of the vectorsassociated with the spatial beams, respective complex delay-domaincombining-coefficients, and a number of indices of the delays associatedwith respective column vectors of a codebook matrix.
 20. A wirelesscommunication network, comprising: at least one receiver, configured toreceive and process a radio signal received via a radio channel from atransmitter employing a plurality of antennas or antenna ports,determine, based on the received signal, complex precoder coefficientsand delays of respective space-delay precoders for each layer andtransmit beam at the transmitter so as to achieve a predefined propertyfor a communication over the radio channel, each space-delay precodercomprising: a double-stage precoding structure, the double-stageprecoding structure including a beamforming matrix that contains PUvectors to form PU spatial beams, U=total number of vectors perpolarization, and P=number of polarizations, where P=1 for co-polarizedantenna arrays at the transmitter and P=2 dual-polarized antenna arraysat the transmitter, wherein the double-stage precoding structure furtherincludes a space-delay-domain combining coefficient vector or matrixincluding complex delay-domain combining-coefficients associated withthe beams and the delays, provide a feedback to the transmitter thedetermined delays and the determined complex precoder coefficients, thecomplex precoder coefficients including the complex delay-domaincombining-coefficients, wherein the feedback includes a precoding matrixidentifier (PMI), the PMI indicating a number of indices of the vectorsassociated with the spatial beams, respective complex delay-domaincombining-coefficients, and a number of indices of the delays associatedwith respective column vectors of a codebook matrix; and at least onetransmitter of claim
 19. 21. The wireless communication network of claim20, wherein the transmitter comprises a base station serving a userequipment, or a user equipment served by a base station, or the receivercomprises a base station serving a user equipment, or a user equipmentserved by a base station.
 22. A method, comprising: receiving andprocessing a radio signal received via a radio channel from atransmitter employing a plurality of antennas or antenna ports,determining, based on the received signal, complex precoder coefficientsand delays of respective space-delay precoders for each layer andtransmit beam at the transmitter so as to achieve a predefined propertyfor a communication over the radio channel, each space-delay precodercomprising: a double-stage precoding structure, the double-stageprecoding structure including a beamforming matrix that contains PUvectors to form PU spatial beams, U=total number of vectors perpolarization, and P=number of polarizations, where P=1 for co-polarizedantenna arrays at the transmitter and P=2 dual-polarized antenna arraysat the transmitter, wherein the double-stage precoding structure furtherincludes a space-delay-domain combining coefficient vector or matrixincluding complex delay-domain combining-coefficients associated withthe beams and the delays, and providing a feedback to the transmitterthe determined delays and the determined complex precoder coefficients,the complex precoder coefficients including the complex delay-domaincombining-coefficients, wherein the feedback includes a precoding matrixidentifier (PMI), the PMI indicating a number of indices of the vectorsassociated with the spatial beams, respective complex delay-domaincombining-coefficients, and a number of indices of the delays associatedwith respective column vectors of a codebook matrix.
 23. Anon-transitory computer program product comprising a computer readablemedium storing instructions which, when executed on a computer, performthe method of claim
 22. 24. A method for forming one or more beams for awireless communication among a transmitter and one or more receivers,the method comprising: applying a set of beamforming weights to one ormore antennas of an antenna array to form the beam, the beam comprisinga transmit beam, wherein the beamforming weights are determinedresponsive to a feedback received from a receiver, the feedbackindicating delays and complex precoder coefficients, the indicateddelays and complex precoder coefficients obtained based on respectivespace-delay precoders for each layer and transmit beam at thetransmitter so as to achieve a predefined property for a communicationover a radio channel to the receiver, the complex precoder coefficientsincluding complex delay-domain combining-coefficients, and eachspace-delay precoder comprising: a double-stage precoding structure, thedouble-stage precoding structure including a beamforming matrix thatcontains PU vectors to form PU spatial beams for the antennas at thetransmitter, U=total number of vectors per polarization, and P=number ofpolarizations, where P=1 for co-polarized antenna arrays at thetransmitter and P=2 dual-polarized antenna arrays at the transmitter,wherein the double-stage precoding structure further includes aspace-delay-domain combining coefficient vector or matrix including thecomplex delay-domain combining-coefficients associated with the beamsand the delays, and wherein the feedback includes a precoding matrixidentifier (PMI), the PMI indicating a number of indices of the vectorsassociated with the spatial beams, respective complex delay-domaincombining-coefficients, and a number of indices of the delays associatedwith respective column vectors of a codebook matrix.
 25. Anon-transitory computer program product comprising a computer readablemedium storing instructions which, when executed on a computer, performthe method of claim 24.